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A    TEXT-BOOK    OF 
THERMODYNAMICS 


A    TEXT-BOOK    OF 

THERMODYNAMICS 

(WITH  SPECIAL  REFERENCE  TO  CHEMISTRY) 


JAMES     RIDDICK     PARTINGTON 

M.Sc. 


With  91   Diagrams 


NEW  YORK 

D.    VAN    NOSTRAND   CO. 

TWENTY-FIVE  PARK  PLACE 

1913 


Ml-' 


PREFACE 

IN  the  following  pages  I  have  endeavoured  to  deduce  the  prin- 
ciples of  Thermodynamics  in  the  simplest  possible  manner  from 
the  two  fundamental  laws,  and  to  illustrate  their  applicability  by 
means  of  a  selection  of  examples.  In  making  the  latter,  I  have 
had  in  view  more  especially  the  requirements  of  students  of 
Physical  Chemistry,  to  whom  the  work  is  addressed.  For  this 
reason  chemical  problems  receive  the  main  consideration,  and 
other  branches  are  either  treated  briefly,  or  (as  in  the  case  of  the 
technical  application  to  steam  and  internal  combustion  engines, 
the  theories  of  radiation,  elasticity,  etc.)  are  not  included  at  all. 
The  arrangement  of  material  adopted  was  the  result  of  careful 
consideration,  and  it  is  hoped  that  the  particular  utility  of  the 
various  methods  in  the  treatment  of  special  problems  has  been 
made  apparent.  It  is  of  course  unlikely  that  this  arrangement 
will  meet  with  agreement  in  all  quarters,  but  it  seemed  to  me  to 
offer  advantages  over  a  strictly  uniform  treatment  in  that  the 
reader  will  thereby  see  more  clearly  the  connection  between  the 
various  methods,  and  hence  his  subsequent  study  of  these  in  the 
original  literature  will  be  rendered  easier.  On  the  one  hand, 
the  student  has  been  informed  by  some  writers  that  the  only 
certain  way  lies  in  the  use  of  the  entropy-function  and  the 
thermodynamic  potentials  ;  on  the  other  hand,  he  is  told  with 
equal  authority  that  the  method  used  by  the  original  investi- 
gators has  been  the  consideration  of  cyclic  processes,  and  that 
the  former  method  is  nothing  but  a  mathematical  (perhaps 
unnecessary)  refinement  of  the  results  obtained  by  the  latter. 
These  extreme  attitudes  appear  to  me  to  be  unfortunate,  and 
more  especially  when  one  observes  the  physical  clearness  intro- 
duced by  the  use  of  cyclic  processes,  but  at  the  same  time 
remembers  that  most  of  the  results  obtained  by  separate  investi- 
gators using  cyclic  processes  had,  with  a  great  many  more, 
previously  been  found  by  J.  Willard  Gibbs  by  means  of  a  purely 
analytical  method. 

The  mathematical  knowledge  pre-supposed  is  limited  to  the 
elements  of  the  differential  and  integral  calculus  ;  for  the  use 
of  those  readers  who  possess  my  Higher  Mathematics  for 


vi  PREFACE 

Chemical  Student*  (Methuen,  1911),  references  to  the  latter  under 
the  symbol  "  H.  M."  have  been  made  whenever  it  appeared 
desirable.  In  spite  of  (or  perhaps  on  account  of)  recent  attempts 
to  prove  the  contrary,  I  am  of  the  opinion  that  no  satisfactory 
progress  can  be  made  even  in  the  elementary  parts  of  thermo- 
dynamics without  a  good  working  knowledge  of  the  calculus. 

In  considering  the  older  literature  of  the  subject,  I  have  not 
thought  it  necessary  to  give  detailed  references,  since  these  will 
be  found  in  any  of  the  standard  text-books  of  physics.  The 
space  thus  made  available  has  been  utilised  in  an  attempt  to 
explain  in  greaterdetail  some  points  which  usually  offer  difficulties 
to  the  student.  The  same  applies  to  the  numerical  constants, 
which  are  to  be  found  in  the  tables  of  Landolt-Bdrnstein,  and  at 
present  are  also  being  actively  revised. 

The  modern  developments  of  the  science  are  considered  in 
the  last  two  chapters.  The  last  chapter  can  be  regarded  only 
as  a  fragmentary  sketch  of  the  subject,  but  a  more  complete 
treatment,  in  view  of  the  limitations  of  space  and  the  very  rapid 
way  in  which  the  theories  are  undergoing  alteration,  appeared 
undesirable. 

In  the  preparation  of  the  book,  most  of  the  existing  treatises 
have  been  consulted,  but  in  the  majority  of  cases  I  have  preferred 
to  use  the  original  literature  of  the  subject. 

All  the  diagrams  were  drawn  specially  for  the  book ;  in  this 
connection  my  best  thanks  are  due  to  Mr.  R.  T.  Hardman,  M.Sc. 
for  valuable  assistance. 

If  the  present  volume  will  help  towards  the  comprehension  of 
the  fundamental  principles  on  which  the  science  of  thermo- 
dynamics rests,  and  also  serve  to  bring  home  the  importance 
of  a  knowledge  of  these  principles  in  the  suggestion  and  inter- 
pretation of  experimental  work,  the  purpose  which  has  been 
kept  in  view  during  its  preparation  will  have  been  amply  fulfilled. 
In  any  case,  it  is  hoped  that  neither  the  extreme  view  that 
thermodynamic  principles  alone  suffice  in  the  construction  of  a 
systematic  physical  or  chemical  science,  nor  the  equally  mistaken 
opinion  that  they  are  of  little  practical  utility  to  the  experimental 
worker,  can  fairly  result  from  its  study. 

J.  R,  PARTINGTON. 
MANCHESTER,  AND  RERUN,  1913. 


TABLE     OF    CONTENTS 


CHAP.  PA3E 

I.       THERMOMETRY    AND    CALORIMETRY  .  .  1 20 

II.       THE      FIRST     LAW     OF     THERMODYNAMICS     AND 

SOME    APPLICATIONS             ....  '21 — 50 

III.       THE       SECOND       LAW       OF        THERMODYNAMICS  ; 

ENTROPY  ......  51 89 

IV.       THE          THERMODYNAMIC  FUNCTIONS          AND 

EQUILIBRIUM.              .....  90 — 116 

V.       FLUIDS 117 — 130 

VI.       IDEAL    AND    PERMANENT    GASES         .              .              .  131 — 108 

VII.       CHANGES    OF    PHYSICAL    STATE            .              .              .  169 — 2*20 

VIII.       VAN  DER  WAALS'  EQUATION    AND    THE    THEORY 

OF   CONTINUITY  OF    STATES         .             .             .  221 — 252 

IX.       THERMOCHEMISTRY          .              .              .              .              .  '253 — 261 

X.       GAS    MIXTURES      ......        262 278 

XI.       THE    ELEMENTARY    THEORY   OF    DILUTE     SOLU- 
TIONS        279—321 

XII.  CHEMICAL    EQUILIBRIUM    IN   GASEOUS    SYSTEMS  322—357 

XIII.  EQUILIBRIUM   IN    DILUTE   SOLUTIONS          .             .  358 — 379 

XIV.  GENERAL     THEORY    OF    MIXTURES     AND      SOLU- 

TIONS       380 — 428 

XV.       CAPILLARITY    AND    ADSORPTION         .  .  .       429 449 

XVI.       ELECTROCHEMISTRY         .....       450 482 

XVH.       THE    THEOREM    OF    NERNST      ....       483 512 

XVffl.       KINETIC    THEORIES    IN   THERMODYNAMICS  .       513 537 

INDEX  539—544 


CORRIGENDA  AND  ADDENDA 

Page  14,  line  2  :  The  method  of  Xernst,  Koref,  and  Lindemann,  by  the  use 
of  the  copper-calorimeter,  determines  the  mean  specific  heat  over 
a  range'  of  temperature.  The  mode  of  procedure  is  the  same  as 
in  ordinary  calorimetry,  except  that  a  hollow  block  of  copper, 
the  temperature  of  which  is  determined  by  means  of  inserted 
thermoelements,  is  used  instead  of  a  calorimetric  liquid,  and  the 
method  therefore  made  applicable  to  very  low  temperatures. 

"ST"  "8T" 

Paye  66,  line  2,  for      -™-    rew.l     jp—  - 

Page  225,  line  2  from  bottom,  for  "  TI  "  read  "  T-»." 

Page  502,  table  of  chemical  constants.  In  the  most  recent  publication 
(Theoretische  Chemie,  7th  edit.,  1913)  Xernst  has  changed  the 
values  slightly  in  some  cases  : 

CO  3-5          H2O  3-6 
C12  3-1          I2      3-9 

The  following  additional  values  may  be  mentioned : 

HI  3-4  SO.  3-3 
NO  3-5  CS,  3-1 
X2O3-3  CeHe  3-0 

Paye  535,  line  1  :   In  Bjerrum's  calculation  the  value  : 

r  =  e/X 

where  c  =  velocity  of  light,  X  =  wave-length  of  the  absorption 
band,  is  substituted  for  »  in  the  Xernst-Lindeniann  equation. 
If  we  put : 


where  /3c  =  14,580,   the  mean  molecular  heats    of    hydrogen, 
oxygen,  and  steam  may  be  expressed  by  the  formulae : 

H2  :  C,  =  E  [2-5  +  <f>  (2-0)] 

02   :  C,  =  R  [2-5  +  0  (2-4)] 
H20  :  Cr  =  B   [  3  +  <»  (3-6)  +  2  0  (1-3)  +  (3^5)'] 


THERMODYNAMICS 

CHAPTER   I 

THERMOMETRY   AND    CALORIMETRY 

1.     Temperature. 

The  volume  of  a  given  quantity  of  air,  or  other  gas,  contained 
in  a  flask  connected  with  a  manometer,  varies  from  day  to  day, 
although  the  barometric  pressures  corresponding  with  two 
unequal  volumes  may  be  equal.  Closer  investigation  discloses 
the  dependence  of  the  volume  of  gas  on  the  degree  of  "  hotness  " 
or  "  coldness  "  of  the  atmosphere,  the  volume  being  greater  on 
hot  days  than  on  cold  days.  The  names  "hot"  and  "cold" 
are  m  general  use  to  denote  specific  sense-impressions;  the 
physical  property  of  the  object  investigated,  on  which  these  sense- 
impressions  depend,  is  called  its  temperature. 

Definition.  -We  shall,  provisionally,  employ  the  word 
'temperature"  to  denote  the  physical  state  of  hotness  or  cold- 
ness of  bodies ;  hot  bodies  are  said  to  have  a  (relatively)  hi^h 
temperature,  cold  bodies  a  (relatively)  low  temperature.  ' 

2.     Measurement  of  Temperature. 

The  measurement  of  temperature  is  achieved  by  making  use  of 
two  experimental  facts : 

(1.)  Certain  states  of  definite  substances  are  marked  off  as 
existing  between  fixed  limits  of  temperature,  and  the  transition 
from  one  state  to  another  occurs,  other  things  being  equal,  at  a 
definite  constant  temperature. 

A  constant  temperature  may  be  recognised  by  unchanging 
volume  in  the  apparatus  of  §  1,  a  rising  or  falling  of  tempera- 
ture by  expansion  or  contraction.  Used  in  this  way  the  apparatus 
may  be  called  a  tiiermoscope. 

The  temperature  of  a  mass  of  ice  rises  on  warming  until  it 


2  THERMODYNAMICS 

reaches  a  point  at  which  the  ice  begins  to  melt.  The  tempera- 
ture then  remains  constant  until  the  whole  of  the  ice  has  melted 
into  water ;  the  latter  then  rises  in  temperature  until  it  begins 
to  boil,  when  its  temperature  again  remains  constant  until  the 
last  drop  has  evaporated.  The  temperature  of  the  steam  then 
rises  progressively.  The  pressure  is  supposed  constant  through- 
out. 

The  constant  temperatures  characterising  the  transitions  : 

Ice           — >  Water 
Water >  Steam 

provide  fixed  points  of  temperature  which  may,  other  things  being 
equal,  be  used  as  points  of  reference.  We  refer  to  them  as  the 
ice-point  and  steam-point,  respectively,  when  the  transitions  occur 
under  normal  atmospheric  pressure  (cf.  §  21). 

(ii.)  Certain  properties  of  definite  substances  change  in  a  con- 
tinuous manner  with  the  temperature,  so  that,  corresponding  with 
each  value  of  the  temperature,  0,  we  have  (it  may  be  within 
certain  limits)  a  definite  value  of  the  property,  x.  If  other 
conditions  are  constant,  a;  is  a  continuous  and  single-valued 
function  of  6 : 

*=/(*) (1) 

Further,  if  the  system,  after  undergoing  such  a  change  of  pro- 
perties, is  brought  back  to  the  initial  temperature,  the  original 
value  of  the  property  is  then  often  regained.  Systems  showing 
the  property  of  reversion  so  defined,  i.e.,  which  exhibit  no 
hysteresis  of  properties  with  respect  to  temperature  variation, 
may  be  used  to  set  up  scales  of  temperature  for  use  in  measure- 
ment, and  so  to  define  temperatures  intermediate  between  the 
fixed  points.  The  interval  is  divided  into  an  arbitrary  number 
of  "  degrees,"  each  corresponding  to  an  equal  increment  of 
the  property,  and  hence  by  definition,  to  an  equal  increment 
of  temperature.  If  the  change  of  volume  of  air  between  the 
ice-point  and  steam-point  is  divided  into  100  equal  parts,  each  of 
these  is  called  a  Centigrade  degree. 

As  examples  of  properties  of  systems  satisfying  the  conditions 
of  definiteness  at  a  particular  temperature  and  of  reversion,  we 
may  refer  to  the  electrical  resistance  of  a  metal  wire ;  the  electro- 
motive force  of  a  thermocouple  with  a  fixed  temperature  at  the  cold 
junction ;  the  volume  of  a  homogeneous  gaseous,  liquid,  or 


THERMOMETRY  AND  CALORIMETRY  3 

solid  substance  ;  the  length  of  a  rod  of  a  solid  substance ;  the 
vapour  pressure  of  a  pure  liquid  such  as  water  ;  the  dissociation 
pressure  of  certain  systems  such  as  nitrogen  tetroxide,  iodine, 
or  ammonium  chloride  (homogeneous),  or  a  salt-hydrate,  a 
carbonate  or  hydride,  etc.  (heterogeneous) ;  and  finally  the 
intensity  of  thermal  radiation  from  a  black  body,  or  hollow 
enclosure. 

3.     Thermometers. 

Any  instrument  which  can  be  used  for  measuring  temperatures 
is  called  a  thermometer.  Thermometers  may  be,  and  are,  con- 
structed which  utilise  any  property  of  a  body  such  as  those 
mentioned  above.  To  evade  the  difficulty  of  comparison  of 
scales,  they  are  usually  all  referred  to  a  gas  thermometer,  with 
Centigrade  scale  as  standard.  The  ice  and  steam-points  on  the 
latter  are  taken  as  0°  and  100°  respectively. 

There  are  in  consequence  various  kinds  of  thermometers,  the 
detailed  description  of  which  belongs  to  books  on  experimental 
physics.  We  may  mention  the  following  : 

(1)  Thermometers  depending  on  volume  (or  pressure)  changes  : 
(i.)  Gas  thermometers,   containing  air,    hydrogen,  or    helium. 

On  account  of  the  low  condensing  temperatures  of 
hydrogen  and  helium,  thermometers  containing  them  are 
particularly  useful  at  very  low  temperatures ;  the  regu- 
larity of  expansion  of  gases  makes  them  very  suitable 
for  thermonietric  purposes. 

(ii.)  Mercury  or  alcohol  thermometers,  used  at  moderate  tempera- 
tures. 

(2)  Electrical  resistance  thermometers,  the  most  widely  used  of 
which  is  Callendars  platinum  resistance  thermometer.     This  is 
probably  the  most  convenient  and  accurate  apparatus  for  measur- 
ing temperatures  between  the  boiling-point  of  liquid  air  ( —  190°  C.) 
and  the  melting-point  of  platinum  (1,500°  C.).  Lead  has  recently 
been  applied  at  very  low  temperatures. 

(3)  Thermocouples  of  platinum  with  an  alloy  of  platinum  and 
10  per  cent,  of  rhodium  or  iridium  are  used  at  higher  tempera- 
tures, and  of  copper  and  constantan  at  lower  temperatures. 

(4)  Optical  pyrometerx,  depending  on    radiation    phenomena, 
may  be  used  up  to  the  highest  attainable  temperatures  (2,500°— 
3,000°  C.). 

B  2 


4  THERMODYNAMICS 

4.  Fundamental  Law  of  Temperature-Reciprocity. 

If  a  hot  body  is  placed  in  contact  with  a  cold  body,  the  tem- 
perature of  the  former  falls,  that  of  the  latter  rises,  until  both 
have  attained  the  same  temperature.  When  the  system  reaches 
a  steady  state,  the  temperature  is  the  same  in  all  its  parts.  The 
change  of  temperature  finds  its  physical  interpretation  in  the 
transfer  of  something  called  heat  from  the  body  of  high  to  the 
body  of  low  temperature,  in  a  similar  manner  to  the  passage  of 
air  from  a  vessel  containing  it  under  high  pressure  to  a  communi- 
cating vessel  containing  it  under  low  pressure.  Tempera- 
ture may  therefore  be  regarded  as  something  analogous  to 
pressure  in  a  fluid,  and  heat  to  the  fluid  itself.  This  analogy, 
in  fact,  constituted  an  earlier  theory  respecting  the  nature  of 
heat,  in  which  it  was  regarded  as  an  "  imponderable  (i.e.,  weight- 
less) fluid  "  and  called  caloric.  The  fact  that  two  bodies  A  and 
C  are  in  thermal  equilibrium  when  they  have  the  same  tempera- 
ture may  appear  to  be  equivalent  to  a  definition  of  temperature. 
If,  however,  the  temperatures  of  A  and  C  have  been  found  to  be 
equal  to  that  of  a  third  body,  B,  used  as  a  thermometer,  they  are 
found  to  have  the  same  temperature  when  tested  by  the  above 
method,  i.e.,  if  A  and  C  are  put  into  direct  contact,  there  is,  in 
the  absence  of  chemical  action,  no  disturbance  of  their  thermal 
states.  Thus,  if  two  bodies  have  equal  temperatures  when  com- 
pared separately  with  a  third  body,  they  are  found  to  have  equal 
temperatures  when  compared  directly  with  each  other. 

This  is  by  no  means  a  self-evident  truth ;  it  is  a  physical  law 
based  on  experience,  and  depends  on  the  property  of  temperature 
equilibrium. 

5.  Unit  of  Heat. 

For  the  purpose  of  measuring  quantities  of  heat  we  require  a 
unit  of  heat,  and  the  preceding  considerations  would  lead  us  to 
define  a  quantity  of  heat  in  terms  of  the  change  of  temperature 
which  it  produces  in  a  given  mass  of  a  particular  substance  under 
specified  conditions. 

Definition. — The  unit  quantity  of  heat  is  that  quantity  of  heat 
which  raises  the  temperature  of  1  gram  of  water  from  15£°  C. 
to  16i°  C.  This  is  called  a  calorie. 

Various  similar  units  have  been  proposed  and  used : 


THERMOMETRY  AND  CALORLMETRY 


Unit. 

Mass  of  Water. 

Rise  of  Temperature. 

(«)  15°  Calorie       . 

1   gram 

16|oC.  >      ,w, 

C. 

(6)  Zero  Calorie     . 

1   gram 

o°  c.  >    P 

C. 

(c)  Mean  Calorie   . 

0-01  gram 

0°    C.  »  1003 

C. 

(rf)  Ostwald's  Calorie  K. 
(e)  4°  Calorie 
(  f)  20=  Calorie       . 

1   gram 
1   gram 
1   gram 

o°  c.  >  IOOD 

3J°  C.  >•      4^: 
19A°  C.  >    20|° 

C. 
C. 
C. 

The  zero  calorie  is  0'06  per  cent,  larger  than  the  15°  calorie, 
whilst  the  mean  calorie  is  0'03  per  cent.  (Behn)  —  0"2  per  cent. 
(Dieterici)  larger  than  the  15°  calorie.  All  temperature's  are 
supposed  to  be  measured  on  the  constant  pressure  hydrogen 
thermometer. 

The  large  number  of  heat  units  employed  by  various  experi- 
menters has  given  rise  to  a  corresponding  amount  of  confusion 
in  the  specification  of  experimental  results,  and  the  name  of  the 
unit  should  now  always  be  given. 

It  is  not  essential,  however,  that  the  unit  of  heat  should  be  defined  in 
terms  of  the  rise  of  temperature  produced  when  heat  is  absorbed  by  a 
standard  body,  say  unit  mass  of  water.  Any  effect  of  heat  absorption 
which  is  capable  of  measurement  and  numerical  expression  might  be  used, 
and  the  method  of  measurement  would  in  all  cases  be  consistent  with  the 
axiom  that  if  two  identical  systems  are  acted  upon  by  heat  in  the  same 
way  so  as  to  produce  two  other  identical  systems,  the  quantities  of  heat 
supplied  to  the  systems  are  equal.  Lavoisier  and  Laplace  (1780-84)  took 
as  unit  that  quantity  of  heat  which  must  be  absorbed  by  unit  mass  of  ice  in 
order  to  convert  it  completely  into  water.  This  unit  is  of  course  different 
from  the  one  we  adopted,  but  if  a  quantity  of  heat  A  has  been  found  to 
raise  from  15^°  to  16^°  twice  as  much  water  as  another  quantity  of  heat  B, 
then  A  will  also  melt  twice  as  much  ice  as  B. 

6.     Heat  Capacity  and  Specific  Heat. 

If  equal  masses  of  lead,  iron,  mercury,  and  glass,  all  having  a 
temperature  of  0°C.,  are  dropped  into  vessels  containing,  say,  100 
grams  of  water  at  50°  C.,  it  will  be  found  that  the  temperature 
falls  in  each  case,  but  when  the  temperatures  have  again  become 
steady  they  are  all  different.  In  accordance  with  our  definition, 
we  say  that  in  each  case  a  certain  number  of  units  of  heat  has 
passed  from  the  water  to  the  body,  but  the  number  so  passing 
before  the  temperature  of  the  body  is  equal  to  that  of  the  (some- 
what cooled)  water  .is  different  for  the  different  substances.  We 


6  THERMODYNAMICS 

may  regard  each  mass  as  having  a  definite  capacity  for  absorbing 
heat.  The  fact  that  the  effect  of  a  given  quantity  of  heat  in 
raising  the  temperature  of  a  body  depends  not  only  on  the  mass 
but  also  on  the  composition  of  the  body  was  clearly  grasped  by 
Joseph  Black  about  1768 ;  this  property  of  bodies  was  named 
Capacity  for  Heat  by  his  pupil  Irvine. 

Definition. — The  heat  capacity  of  a  body,  under  specified  condi- 
tions, is  measured  by  the  number  of  heat  units  which  must  pass 
into  that  body  to  raise  its  temperature  1°  C. 

It  has  been  shown  experimentally  that  if  a  body  or  system  in 
a  given  state  A  is  converted  into  another  B,  by  the  absorption  of 
a  definite  quantity  of  heat,  the  same  amount  of  heat  is  given  out 
again  if  B  is  reconverted  into  A,  so  that  all  the  intermediate  states 
between  A  and  B  are  retraced  in  the  same  order  as  they  occurred  in 
the  first  operation. 

In  this  connexion  the  fluid  analogy  may  be  useful  in  giving  precision  to 
our  ideas.  It  is  evident  that  very  different  volumes  of  water  must  be  poured 
into  cylinders  of  different  diameter  in  order  that  the  level  of  water  may  be 
the  same  in  all.  The  rise  of  level  is  proportional  to  the  volume  of  water 
poured  in,  and  inversely  proportional  to  the  area  of  cross-section  of  the 
cylinder.  If  quantities  of  water  are  poured  into  the  cylinders  so  as  to 
produce  unit  rise  of  level  in  all,  the  volumes  will  be  proportional  to  the 
cross-sections,  and  the  volumes  of  water  may  be  used  as  measures  of  the 
latter,  which  obviously  correspond  with  the  "capacities"  of  the  jars  for 
holding  liquid.  In  the  same  way  we  have  defined  the  capacity  of  a  body 
for  absorbing  heat  as  measured  by  the  number  of  heat  units  which  must 
be  put  into  the  body  in  order  to  raise  its  temperature,  or  heat  level,  by  one 
unit.  We  must  remember,  however,  that  heat  capacity  is  not  heat,  any 
more  than  bulk  capacity  is  water  used  in  measuring  it.  Although  the  heat 
capacity  of  a  body  is  measured  in  terms  of  the  number  of  heat  units  required 
to  produce  a  standard  change  of  one  property  of  the  body,  it  denotes  a 
specific  property  of  the  body  itself.  The  use  of  the  fluid  analogy  obviously 
does  not  affect  the  truth  or  otherwise  of  any  equations  derived  from  the 
definition  of  heat  capacity,  because  the  latter  involves  only  measure- 
ments of  temperatures  and  masses,  magnitudes  directly  ascertainable 
by  experiment  and  independent  of  any  theoretical  views  on  the  nature  of 
heat. 

If  Q  units  of  heat  are  required  to  raise  the  temperature  of  a 
body  1°,  then  2Q  will  be  required  to  raise  the  temperature  of  two 
such  bodies  through  1°,  and  so  on.  Hence  the  heat  capacity 
of  a  homogeneous  body  is  proportional  to  its  mass.  The  heat 
capacity  of  unit  mass  of  a  homogeneous  body  may  therefore  be 


THERMOMETRY  AND  CALORIMETRY  7 

regarded  as  measuring  some  specific  property  of  the  substance  of 
which  the  body  is  composed. 

Definition. — The  heat  capacity  of  unit  mass  of  a  substance  is 
called  its  specific  heat. 

The  name  "  specific  heat  "  was  introduced  by  Gadolin  in  1784. 

As  unit  mass  we  shall  take  1  gram,  and  unit  rise  of 
temperature  1°  C.  on  the  hydrogen  gas  thermometer. 

The  number  of  heat  units  which  must  be  imparted  to  a  mass 
m  of  a  substance  to  raise  its  temperature  through  1°  under 
specified  conditions  will  be  : 

Q  =  me (1) 

where  c  is  the  heat  capacity  of  unit  mass,  or  the  specific  heat,  of 
the  substance,  under  the  given  conditions. 

The  specific  heat  of  a  substance  must  always  be  defined  relatively  to  a 
particular  set  of  conditions  under  which  heat  is  imparted,  and  it  is  here  that 
the  fluid  analogy  is  very  liable  to  lead  to  error.  The  number  of  heat  units 
required  to  produce  unit  rise  of  temperature  in  a  body  depends  in  fact  on 
the  manner  in  which  the  heat  is  communicated.  In  particular,  it  is  dif- 
ferent according  as  the  volume  or  the  pressure  is  kept  constant  during  the 
rise  of  temperature,  and  we  have  to  distinguish  between  specific  heats  (and 
also  heat  capacities)  at  constant  volume  and  those  at  constant  pressure,  as 
well  as  other  kinds  to  be  considered  later. 

If  the  temperature  of  a  mass  m  of  a  substance  rises  from  61  to 
02,  we  may  represent  the  amount  of  heat  absorbed  by : 

Q  =  me  fa  -  00    .         .         .         .     (2) 

if  we  understand  by  c  the  average  heat  capacity  of  unit  mass  in 
the  range  of  temperature  considered.  It  must,  however,  be 
carefully  noted  that  the  amounts  of  heat  required  to  raise  the 
temperature  of  unit  mass  of  a  substance  from  0°  to  1°,  from  1°  to 
2°,  and  so  on,  are  not  usually  equal,  because  after  the  temperature 
has  been  altered  from  0°  to  1°  we  are  not  dealing  with  the  same 
substance  as  we  started  with,  but  with  a  warmer  substance,  and 
it  is  quite  possible  that  the  specific  heat  of  the  latter  is  different 
from  that  of  the  initial  substance.  In  many  cases,  however,  c  is 
nearly  independent  of  temperature  over  a  range  of  a  few  degrees, 
so  that  we  may  assume  that  the  same  number  of  calories  will  be 
required  for  the  following  changes  of  temperature : 

0,  to  (0x  +  1),  (*i  +  1)  to  (0!  +  2),  .  .  .  (02  -  1)  to  0.2, 
provided  (02  —  0i)  is  not  large.     The  admissible  range  of  tempera- 


8  THERMODYNAMICS 

ture  varies  very  much  with  the  nature  of  the  substance  and  with 
the  initial  temperature. 

The  variation  of  specific  heat  with  temperature  was  discovered 
by  Dulong  and  Petit  in  1819.  It  explains  why  so  many  different 
heat  units  exist  (cf.  §  5),  and  requires  the  definition  of  specific 
heat  to  be  so  framed  as  to  allow  for  this  variation.  For  this 
purpose  we  replace  the  finite  changes  by  infinitesimal  ones.  If 
SQ  units  of  heat  are  absorbed  when  unit  mass  of  a  substance  is 
raised  in  temperature  from  (6—  |  80)  to  (6-\-\  80)  under  specified 
conditions,  the  true  specific  heat  at  the  temperature  6  is  : 


It  must  be  observed  that  the  quotient  has  a  definite  limiting 
value  only  when  the  conditions  of  heating  (e.g.,  constant  pressure) 
are  specified. 

The  heat  Q  absorbed  in  a  finite  rise  of  temperature  from  0i  to 


%d0  =    mdo  .     .  (4) 


The  form  of  /(0)  can  only  be  determined  by  experiment,  or  by 
means  of  atomistic  hypotheses  (cf.  Chap.  XVIII.).  Since  c6  is 
known  to  change  continuously  and  only  comparatively  slowly 
with  rise  of  temperature,  we  may  expand  /(0)  in  a  Maclaurin's 
series  (cf.  H.  M.,  §  95)  : 


or  ce  =c0  +  a<9  +  Z>02+  .          .         .         .     (5) 

where  c0  =/(0)  =  specific  heat  for  6  =  0. 

Thus,  if  we  take  only  terms  as  far  as  02  into  account  (which  is 
usually  abundantly  sufficient)  : 


.     (6) 
.  (6a) 


THERMOMETRY  AND   CALORIMETRY 

If  we  compare  equations  (2)  and  (4)  we  find  (m  =  1) : 


.  ,     .         ,         .     (7) 

so  that  the  mean  specific  heat  in  the  interval  (02  —  #1)  is  the  mean 
value  of  the  true  specific  heat  in  the  same  interval  (cf.  H.  M., 
§  H7). 

Corollaries. — (1)  To  find  the  true  specific  heat  when  the  mean 
specific  heat  is  given,  multiply  the  latter  by  0  and  differentiate 
with  respect  to  0. 

(2)  If  c»  =  Co  +  a0  +  b02 

then         c  =  cv  +  |  (6l  +  6>2)  +  |  (<V  +  0A  +  022) 

(3)  If   the   zero  of  temperature    is    transferred    to    a    point 
—  273°  C.,  so  that,  on  the  new  scale  : 

T  =  0  +  273 

dT  =  d  (0  +  273)  =  ae 

dQ  =  ced  (0  +  273)  =  ced9  =  cgdT 

show  that  the  mean  specific  heat  between  0°  and  T°  and  that 
between  0°  C.  and  0°  C.  are : 


where  ce  =  c0  +  a0,  cT  =  c0  +  a(T  —  273)  =  (c0—  273a)  +  aT  = 
c'0  +  «T,  are  the  true  specific  heats. 

7.     The  Specific  Heats  of  Gases. 

We  have  to  distinguish,  in  the  case  of  gases,  between  : 

(i.)  The  specific  heat  at  constant  pressure,  denoted  by  cp>  and 

(ii.)  The  specific  heat  at  constant  volume,  denoted  by  cv_ 

The  first  accurate  measurements  of  the  specific  heats  of  gases  at  constant 
pressure  were  made  by  Eegnault  (1862),  who  used  the  "constant  flow" 
method.  A  slow  stream,  of  gas  from  a  large  reservoir  was  allowed  to  pass 
through  a  pressure  regulator,  and  then  through  a  copper  spiral  immersed 
in  an  oil  bath  of  known  temperature.  The  hot  gas  then  passed  through 


10  THERMODYNAMICS 

a  series  of  thin  metal  boxes  immersed  in  a  water  calorimeter,  to  which  it 
gave  up  heat. 

E.  Wiedemann  (1876)  replaced  the  heating  coil  and  metal  boxes  by 
metal  tubes  filled  with  metal  turnings,  thus  exposing  a  larger  surface  to  the 
gas. 

Holborn  and  Austin  (1905)  have  recently  extended  the  method  so  as  to 
make  it  applicable  at  high  temperatures.  The  gas  was  sent  through  a 
heater  consisting  of  a  narrow,  electrically  heated,  nickel  tube  containing 
nickel  turnings,  and  then  passed  into  a  silver  calorimeter  containing  water. 
Initial  temperatures  up  to  800°  were  used.  The  method  was  also  used  by 
Holborn  and  Henning  (1905)  to  determine  the  specific  heat  of  steam  at  high 
temperatures. 

The  measurement  of  the  specific  heat  at  constant  volume  is  attended  with 
considerable  difficulty,  because  the  thermal  capacity  of  a  vessel  strong  enough 
to  contain  the  gas  after  heating  has  a  value  much  greater  than  that  of  the 
thermal  capacity  of  the  enclosed  gas. 

Mallard  and  Le  Chatelier  have  measured  the  specific  heats  at  constant 
volume  at  high  temperatures  by  their  so-called  explosion  method.  In  this  an 
explosive  mixture  (e.g.,  2CO  +  02)  is  fired  in  a  very  strong  metal  chamber, 
and  the  pressure  measured  by  an  indicator  similar  to  those  used  in  recording 
the  pressures  attained  in  engine  cylinders.  By  assuming  that  the  gases 
conformed  to  the  laws  of  Boyle  and  Charles,  the  mean  specific  heats  between 
the  ordinary  temperature  and  the  explosion  temperature  could  be  calculated. 
Berthelot  and  Vieille  (1882-5),  instead  of  measuring  the  static  pressure  as 
above,  allowed  the  force  of  the  explosion  to  deform  a  copper  cylinder,  placed 
beneath  a  steel  piston  exposed  to  the  gas  ("  crusher  manometer  ").  Langen 
(1903)  and  M.  Pier  (1908)  have  made  the  most  recent  determinations  with  the 
explosion  method.  Sarrau  and  Vieille  (1882-7)  used  the  crusher  manometer, 
but  instead  of  firing  an  explosive  gas  mixture,  they  ignited  a  charge  of  high 
explosive  hanging  freely  in  the  simple  gas. 

Measurements  of  cv  at  moderate  temperatures  were  made  by  Joly  in  a 
steam  calorimeter.  The  weight  of  water  condensed  from  steam  blown 
through  a  chamber  containing  a  copper  globe  filled  with  gas  was  compared 
with  the  weight  deposited  on  a  similar  but  vacuous  globe. 

Eegnault  and  Wiedemann,  cf.  Haber's  Thermodynamics  of  Technical  Gas 
Reactions,  Eng.  trans.,  lect.  6. 

Holborn  and  Austin,  Sitzungsber.  Konigl.  preuss.  Akad.  (1905),  p.  175. 

Holborn  and  Henning,  Ann.  der  Phys.,  [4],  18,  739  (1905). 

Mallard  and  Le  Chatelier,  Journ.  de  Phys.,  [2],  1,  173  (1882) ;  Ann.  des 
Mines,  [8],  4,  274  (1883). 

Berthelot  and  Vieille,  C.  E.,  95,  1280  (1882);  96,  116,  1218,  1358  (1883); 
Ann.  chim.  phys.,  4,  13  (1885). 

Langen,  Mitteilungen  uber  Forschungsarbeiten  aus  dem  Geibiete  des 
Ingenieurwesens,  Berlin,  1903,  Heft  8. 

M.  Pier,  Zeitschr.  physical.  Chem.,  62,  397  (1908)  ;  66,  759  (1909) ; 
Zeitschr.  Ekklrochem.,  15,  536  (1909). 

N.  Bjerrum,  Zeitschr.  Elektrochem.,  17  (1911) ;  Zeitschr.  phi/sikal.  Chem.,  79, 
513,  537  (1912). 


THERMOMETRY  AND   CALORIMETRY  11 

Sarrau  and  Vieille,  C.  R.,  95,  26,  133,  181  (1882);  102,  1054  (1886);  104, 
1759  (1887).  Cf.  Heise,  Sprengstojfe  und  Zundung  der  Sprengschiisse, 
Berlin,  1904. 

Joly.  Proc.  Boy.  Soc.,  55,  390  (1894);  Phil.  Trans.,  182,  73  (1892). 

Measurements  of  the  effect  of  pressure  on  cp  have  been  made  by  Lussana 
(1895  —  8).  With  pressures  of  5  —  150  atm.  he  found  cp  to  increase  consider- 
ably with  the  pressure  with  all  the  gases  investigated.  In  the  case  of  air 
and  carbon  dioxide  the  results  showed  that  cp  would  reach  a  maximum 
and  then  decrease  at  higher  pressures  —  a  result  directly  verified  in  the  case 
of  air  by  Witkowski  (1895).  Lussana's  results  are  represented  by: 


H2 

=  (cp  )iatm.   + 
(cp  )iatm. 
3-4025 

»(P-  !)• 
0-013300 

OH4 

C02 

0-5915 
0-2013 

0-003463 
0-0019199 

C2H, 

,         0-40387 

0-0016022 

N20 

0-22480 

0-0018364 

Air  :  cp  =  0-23702  +  0-0015504  (p  —  1)  —  0-00000195  (p  —  I)2. 

The  more  closely  a  gas  approaches  its  point  of  liquefaction  the  greater  is 
the  influence  of  pressure  on  its  specific  heat. 

The  effect  of  pressure  on  the  specific  heat  of  steam  has  been  examined  by 
Thiesen  and  by  Lorenz.  The  molecular  heat  has  a  minimum,  Cp  =  7-34,  at 
80°  C.  With  diminishing  pressure,  steam  behaves  more  and  more  like  a 
permanent  gas,  the  molecular  heat  tending  to  a  limiting  value  Cp  =  7  -74. 

Lussana,  Nouvo  dm.,  [3],  36,  5,  70,  130  (1894);  Fortschr.  der  Phijs.  (1896), 
345;  (1897),  331  ;  Witkowski,  BuU.  internat.  de  I'Acad.  Sci.  Cracovie  (1895), 
290;  Fortschr.  der  Phys.  (1896),  343;  cf.  Amagat,  C.  E.,  122,  66,  121  (1896); 
Thiesen,  Ann.  der  Phijs.,  [4],  9,  88  (1902);  Tumlicz,  Wein.  Akad.  Her. 
(1897),  Ha,  654;  (1899),  Ha,  1395;  Macfarlane  Gray,  Phil.  Mag.  (1882), 
13,337;  Amagat,  C.  R.,  142,1120,1303  (1906);  143,  6  (1906);  Journ.de 
Phys.,  [3],  9,  417  (1900);  [4],  5,  637  (1906);  Dalton,  Phil.  Mag.,  [6],  13, 
536  (1907)  ;  Nernst,  Vtrhl.  d.  d.  Phys.  Ges.,  11,  320  (1909). 

According  to  Haber  (loc.  (.it.,  p.  131)  the  results  of  Langen  are  probably 
correct  to  3  per  cent,  even  at  2,000°.  Pier,  in  his  recent  explosion  experi- 
ments, has  shown,  however,  that  the  maximum  pressures  were  not  obtained 
by  the  previous  observers,  on  account  of  the  oscillations  of  their  manometers, 
He  used  a  steel  plate  with  very  high  frequency  of  vibration,  and  registered 
the  distortion  by  reflecting  a  beam  of  light  from  a  mirror  attached  to  the 
manometer  disc  on  to  a  revolving  drum  of  sensitised  paper.  The  recorded 
curves  show  a  well-defined  maximum  pressure,  and  his  results  are  probably 
accurate  to  1  per  cent.  Values  of  C,,  : 

Argon  2-98    constant  to  2,350°  C. 

N2  4-900  +  0-000450  (1,300°—  2,500°) 

H2  4  -'7  00  -4-  0-000456 

HC1  4-600  +  0-0050 

Steam  6'065  +  0'00050  +  0-2  X  10-»03 


12  THERMODYNAMICS 

Of.  also  F.  Keutel,  Inauy.  Dies.,  Berlin,  1910  ;  E.  Thibaut,  Inaug.  Dies., 
Berlin,  1910;  Petrini,  Zeitschr.  physik.  Chem.,  16,  97.  A  very  complete 
account  of  the  data  up  to  1908  will.be  found  in  the  work  of  Haber  cited 
above. 

8.     The  Specific  Heats  of  Vapours. 

The  results  of  Regnault  and  of  E.  Wiedemann  (which  are  not 
in  good  agreement)  indicate  that  : 

(1)  The  specific  heats  of  vapours  run  parallel  to  those  of  the 
liquids,  and  the  temperature  coefficients  a  : 


are  only  slightly  different  in  the  two  cases. 

cp  vapour  at,  0°.      cv  liq.  at  0°.  a  vap.  a  liq. 

CHC13  .    .  0-1341    0-2323    0-000135    O'OOOlOl 
C6H6   .    .  0-2237    0-3798    0*00102     0*00144 
(C2H5)20    .  0-3725    0-5290    0'00085     0-00059 

cv  X  molecular  wt. 

(2)  The   quotient  -  —  =  -  ;  -  r     is     not    constant, 

no.  of  atoms  in  molecule 

varying  between  2*0  and  4*5,  and  depending  on  the  temperature. 

(3)  The  specific  heats  increase  with  rise  of  temperature    to 
different  extents  with  different  vapours.     (Cf  .  the  work  of  Thibaut 
quoted  in  §  7.) 

9.     The  Specific  Heats  of  Solids. 

The  variation  of  specific  heat  of  a  solid  with  temperature  was, 
as  has  already  been  stated,  discovered  by  Dulong  and  Petit  as 
early  as  1819,  but  it  is  only  quite  recently  that  the  remarkable 
relations  exhibited  at  very  low  temperatures  have  come  to 
light  (cf.  Chap.  XVIIL). 

In  most  cases  ce  is,  at  the  ordinary  temperature,  approximately 
a  linear  function  of  6  : 

c  iron  (15°  to  320°)  =  0'10442  (1  -f  -001290). 

The  specific  heats  of  solids  at  low  temperatures  are  appre- 
ciably less  than  at  higher  temperatures.  A  maximum  specific 
heat  has  been  observed  in  the  case  of  iron  at  740°  and  nickel  at 
320°  (Lecher,  1908).  Since  these  are  the  temperatures  at  which 
recalescence  and  loss  of  magnetic  properties  occur,  the  close 
relation  of  specific  heat  to  molecular  structure  is  evident. 


THERMOMETRY  AND  CALORIMETRY  13 

The  most   marked  effect  of   change  of   temperature   on   the 
specific  heat  is,  however,  exhibited  by  the  non-metals,  carbon, 
boron,   and   silicon.       The   following  values  were   obtained  by 
H.F.Weber  (1875): 
Diamond  (  —  50°  to  +  250°)  : 

c$  =  '0947  +  '0009940  —  -0000003603 
Graphite  (0°) :  ce  =  152  +  '00070. 
Boron  (0°  to  250°):  ce  =  0"22  +  0'000710. 
Silicon  (cryst.)  : 

0=         -  50°        0°         +  50°         100°        200° 
c9=  0-13         -16  -18         "195         -202 

At  higher  temperatures  than  those  over  which  the  formulae 
hold  good  the  rate  of  increase  of  the  specific  heat  with  tempera- 
ture falls  off  rapidly,  and  the  specific  heat  tends  to  a  limiting 
value  which  changes  only  slightly  with  further  increase  of 
temperature.  These  limiting  values  are  (approximately)  : 

Diamond       0'46  Boron       0'50 

Graphite      0'47  Silicon      0'205 

At  the  other  temperature  extreme  we  have  the  measurements 
of  specific  heats  executed  at  the  temperatures  of  liquefied  gases. 
A  known  mass  of  the  substance  is  dropped  into  liquid  carbon 
dioxide  (—  78°),  oxygen  (-  183°),  or  hydrogen  (-  250°),  and 
the  volume  of  gas  liberated  is  measured. 

Dewar,  Proc.  Roy.  Soc.,  A,  76,  325  (1905),  finds  the  following 
values  of  the  specific  heats  : 

18°  to  -78°  -78°  to  -188°  -  188°  to  -  252.i° 

Diamond          0*0794  0'0190  0*0043 

Graphite          01341  0'0599  0*0133 

Ice  0-463(-18°to-78°)  0-285  0146 

The  specific  heats  of  diamond  and  graphite  are  reduced  to  ^  and 
-j\y  respectively  between  the  ordinary  temperature  and  the  boiling- 
point  of  liquid  hydrogen ;  the  specific  heats  of  the  substances 
between  the  temperatures  of  liquid  air  and  liquid  hydrogen  are  in 
fact  less  than  those  of  any  other  substances,  even  less  than  that 
of  a  gas  at  constant  volume. 


14  THERMODYNAMICS 

The  method  just  described  leads  to  the  mean  specific  heats  over 
a  fairly  large  range.  Nernst,  Koref,  and  Lindemann  (1910)  have 
recently  described  a  method  of  measuring  the  true  specific  heat  at 
a  given  low  temperature.  The  substance  is  contained  in  a  block 
of  copper  cooled  to  the  requisite  temperature  in  liquid  carbon 
dioxide,  liquid  air,  etc.,  and  energy  is  supplied  by  a  heating  spiral 
of  platinum  wire  carrying  an  electric  current,  the  measurement 
of  the  resistance  of  which  serves  at  the  same  time  to  determine 
the  temperature. 

Measurements  of  the  true  specific  heats  at  low  temperatures 
have  been  carried  out  by  Eucken,  who  worked  in  such  a  way 
that  to  a  weighed  quantity  of  the  substance  in  the  form  of  a 
block,  or  in  a  proper  isolated  vessel,  a  known  quantity  of  heat 
was  added  by  means  of  an  electrically  heated  platinum  spiral, 
the  resistance  of  which  at  the  same  time  served  to  measure  the 
temperature.  The  quotient  of  the  electrical  energy  spent  by  the 
rise  of  temperature  gives  the  specific  heat.  The  correction  for 
cooling  (in  vacuum  of  ^Q  mm-)  amounted  to  20  per  cent.,  and  for 
heat  capacity  of  the  apparatus  5  per  cent.,  yet  the  results  are 
stated  to  be  accurate  to  1  per  cent. 

For  very  low  temperatures  a  lead  wire  is  used  as  thermo- 
meter, with  a  heating  coil  of  constantan  wire  (Kamerlingh 
Onnes). 

The  method  has  been  applied  by  Eucken  to  determine  specific 
heats  of  gases  (e.g.,  H2)  at  constant  volume  by  enclosing  them  in 
small  metallic  vessels. 

The  following  references  include  most  of  the  recent  experi- 
mental data  : 

Eucken,  Physik.  Zeitschr.,  10,  586  (1909)  ;  Sitzungsberichte  Kgl. 
Akad.,  Berlin  (Berl.  Ber.)  (1912),  p.  141  ;  Fortschritte  dcrClim/., 
[iv.],  105  (1911). 

Lindemann,  Berl  Ber.  (1910),  12,  247 ;  13, 316;  (1911),  22,  492; 
Dissertation,  Berlin,  1911;  Lindemann  and  Nernst,  Zeitxcltr. 
Elektrochem.,  18,  817  (1911). 

Koref,  Berl.  Ber.  (1910),  12,  253  ;  Ann.  Plnjs.,  [iv.],  36,  49 
(1911). 

Nernst,  Berl.  Ber.  (1910),  12,  247,  262  ;  13, 306  ;  (1911),  22, 492; 
Ann.  Phys.,  [iv.],  36,  395  (1911). 

Magnus,  Zeitschr.  Elektrochem.,  16, 269  (1910) ;  Ann.  Phys.,  [iv.], 
31,  597  (1910). 


THERMOMETRY  AND   CALORIMETRY 


15 


Pollitzer,  Zeitschr.  Elektrochem.,  17,5  (1911). 

Gaede,  Dissertation,  Freiburg,  1902. 

Schimpff,  Zeitschr.  physik.  Chem.,  71,  257  (1910). 

Wigand,  Ann.  Plujs.,  [iv.],  22,  79  (1907). 

Russell,  Physik.  Zeitschr.,  13,  59  (1912). 

Kamerlingh  Onnes,  Commun.  Phys.  Lab.  Leiden,  No.  119, 
1911. 

Dependence  on  Density. — If  the  density  of  a  metal  is 
increased  by  hammering,  its  specific  heat  is  slightly  decreased. 
The  same  change  is  observed. if  the  change  of  density  is  due  to  a 
change  of  crystalline  form,  or  to  change  from  an  amorphous 
state  to  a  crystalline  state,  and  with  different  allotropic  forms 
(Wigand,  loc.  cit). 


Density. 

Sp.  Ht. 

Range. 

i  Diamond    . 

3-518 

0-1128 

10-7° 

Carbon 

Graphite     . 
(  Gas  carbon 

2-25 
1-885 

•1604 
•2040 

10-8° 
24°  >68° 

/  rhombic 

2-06 

•1728 

0°  >54° 

„  .  , 

\  inonoclinic  . 

1-96 

•1809 

0°  >52° 

bulpnur 

j  amorph.  (insol.) 

1-89 

•1902 

0°  >53° 

,,        (sol.) 

1-86 

•2483 

0°  >50° 

Tin 

(  White 
{  Grey  .        . 

7-30 
5-85 

•0542 
•0589 

0°  >21° 

According  to  van't  Hoff  (1904),  the  form  which  is  stable  at 
high  temperatures  has  the  greatest  specific  heat ;  this  rule  has 
some  exceptions.  (Cf.  also  Griineisen,  Ann.  Phys.,  [iv.],  26,  393 
(1908).  ) 

Relation  of  Specific  Heat  to  Atomic  Weight  in  the  Case  of  Solids. 

(a)  Dulong  and  Petit  in  1819  arrived  at  the  very  remarkable 
experimental  law  that  the  product 

(At.  Wt.)  X  (Sp.  Ht.)  =  ac  =  atomic  heat 

is  approximately  constant  at  the  ordinary  temperature  for  solid 
elements  of  atomic  weight  exceeding  30,  and  is  approximately  6'4. 

The  heat  capacities  of  a  gram-atom  of  these  various  elements 
are  therefore  very  nearly  equal. 

The  order  of  approximation  which  obtains  may  be  seen  from 
the  following  figures  (A.  Magnus,  1910) : 


16 


THERMODYNAMICS 


Element. 

Range  of  Temperature. 

Atomic  Hc;it. 

Lead  J 

18°  >100°  C. 
16°  »256° 

6-409 
6-606 

( 

16°  >100° 

5-570 

Aluminium 

16°  »304° 

6-097 

( 

17°  »545° 

6-475 

Copper        .... 

15°  >238° 
15°  >338° 

6-048 
6-090 

(/3)  The  researches  of  F.  Neumann  (1831),  Regnault  (1840), 
and   H.    Kopp   (1864),   indicated   that   solid   elements  preserve 
unchanged  their  atomic  heats  when  they  unite  to  form  solid 
compounds.      Thus,  the   product   (molecular  weight}  X  (specific 
heat)  =  (molecular  heat)  is  composed  additively  of  the  atomic  heats  : 
MC  =  niajCi  +  «2«2C2  +  n3a3cs  +         .         .    (9) 
where   MC  =  molecular,weight  and  specific  heat  of  compound 
«!,  a2  =  atomic  weights,  and 

d,  cz  =  specific  heats,  of  the  elements  present  in  the 
molecule,  the  numbers  of  gram-atoms  being  n\,  «2,  per  gram 
molecule  of  the  compound. 

The  law  was  stated  in  this  form  by  J.  P.  Joule  in  1844  ;  it  is 
usually  referred  to  as  Woestyn's  law  (1848).  It  shows  that  the 
carriers  of  heat  in  a  solid  compound  are  not  the  molecules  of  the 
latter,  but  the  atoms  of  its  constituent  elements.  Joule's  law 
enables  one  to  calculate  the  molecular  heats  of  compounds  from 
the  atomic  heats  of  their  elements,  and  the  atomic  heats  of 
elements  in  the  solid  state  when  the  latter  are  not  readily  directly 
accessible  (e.g.,  solid  oxygen,  from :  c(CaC03)  —  c(Ca)  —  c(C)  = 
3c(0),  or  100  X  0-203  —  6'4  —  1-8  =  3  X  4'0). 

Specific  Heats  of  Solid  Mixtures. — The  specific  heat  of  a 
homogeneous  solid  mixture  of  solid  components  is  not  usually 
additively  composed  of  the  specific  heats  of  the  latter.  W.  Spring 
(1886)  found  that  the  total  heat  capacity  of  alloys  of  lead  and  tin 
was  always  greater  than  the  sum  of  those  of  the  components,  but 
above  the  melting-point  the  two  were  equal.  A.  Bogojawlensky 
and  N.  Winogradoff  (1908)  find,  however,  that  the  heat  capacities 
of  the  isomorphous  mixtures  : 

?n-chloro-  +  ?n-bromo-nitrobenzene 
a-chloro-  +  a-bromo-cinnamaldehyde 
azobenzene  +  dibenzyl 


THERMOMETEY  AND   CALORIMETRY  17 

are,  both  in  the  liquid  and  solid  state,  composed  additively  of  the 
heat  capacities  of  the  constituents. 

10.    The  Specific  Heats  of  Liquids. 

As  no  simple  relations  between  the  specific  heats  of  liquids 
have  yet  been  arrived  at,  we  shall  merely  bring  together  a  few 
generalisations  from  the  experimental  data  : 

(a)  The  specific  heat  of  the  solid  modification  is  usually  less 
than  that  of  the  liquid : 

Solid.          Liquid.       Gaseous. 
H20  -504  1-000  -477 

Hg  '0319  -0333 

(j3)  The  specific  heats  of  liquids  depend  on  the  temperature, 
usually  increasing,  but  sometimes  decreasing,  with  rise  of 
temperature. 

(y)  v.  Reis  (1887)  found  that  the  differences  between  the 
products  (sp.  ht.  between  0°  and  boiling-pt.)  X  (mol.  ict.}  were 
nearly  constant  in  particular  homologous  series  of  organic 
liquids. 

R.  Schiff  (1886)  found  some  simple  relations  between  specific 
heat  and  composition  in  organic  liquids. 

(8)  Specific  Heats  of  Mixtures  of  Liquids. — With  the  exception 
of  the  cases  of  liquid  alloys,  and  some  fused  isomorphous  sub- 
stances, the  heat  capacity  of  a  mixture  of  liquids  is  usually 
greater  than  the  sum  of  the  heat  capacities  of  the  components 
(Bussy  and  Buignet,  1865).  This  is  the  case  with  mixtures  of 
alcohol  with  H20,  CHC13,  CS2,  C6H6;  the  20  per  cent,  solution  of 
alcohol  in  water  having  a  specific  heat  (1'046)  greater  than  that 
of  any  other  liquid  below  100°.  On  the  other  hand,  mixtures  of 
C6H6  and  CHC13  have  heat  capacities  the  sum  of  those  of  the 
constituents  (J.  H.  Schiiller,  1871).  According  to  Ramsay  and 
Shields,  H20  and  C2H5OH  are  associated,  C6H6  and  CHC13  are 
not.  Further  investigation  with  other  liquids  would  certainly 
lead  to  interesting  results,  if  these  were  taken  in  connexion  with 
the  heat  of  admixture,  and  the  other  properties  of  the  components 
and  mixture. 

(e)  Specific  Heats  of  Solutions. — Dilute  aqueous  solutions  of 
many  salts  exhibit  a  peculiarity  which  has  not  yet  been  satis- 
factorily explained.  Let  p  gr.  salt  be  dissolved  in  100  gr.  water ; 

T.  c 


18  THEBMODYNAMICS 

if  c  is  the  specific  heat  of  the  solution,  c(100  +  p)  is  the  weight 
of  water  which  has  the  same  heat  capacity  as  (100  +  p)  gr.  of 
solution  ("  water-equivalent  "  of  the  solution).  Now  c(100  -f-  p) 
is  frequently  less  than  100,  i.e.,  the  total  heat  capacity  of  the 
solution  is  less  than  that  of  the  water  alone  contained  in  it. 
With  sodium  chloride,  for  example,  the  numbers  are  : 

p  5  10  20  30 

A  =  100  —  c(100  +  p)        2-29         2-01         0'36         —  2'66. 
Thus,  c(100  +  p)  <  100  at  small  concentrations, 

=  100  at  medium  concentrations, 
>  100  at  high  concentrations. 

At  very  great  dilution  A  approaches  a  limiting  value  (Schiiller, 
1869  ;  Thomsen,  1870  ;  Marignac,  1871-6). 

11.     Latent  Heat. 

Let  a  quantity  of  powdered  ice  be  placed  in  a  vessel  along  with 
a  thermometer,  and  the  whole  placed  in  hot  water.  The  tempera- 
ture of  the  latter  falls,  showing  that  heat  is  being  absorbed  from 
it,  but  the  temperatures  of  the  ice,  and  the  water  formed  by  its 
fusion,  remain  constant  (provided  the  mass  is  efficiently  stirred) 
until  the  last  portion  of  ice  has  melted.  The  result  of  absorption 
of  heat  by  ice  is  therefore  the  production  of  water  without  any 
change  of  temperature.  In  the  same  way,  the  application  of  heat 
to  water  at  its  boiling-point  gives  rise  to  the  production  of  steam 
at  the  same  temperature. 

There  are  many  other  examples  of  changes  in  which  a  solid 
passes  into  a  liquid,  or  a  liquid  into  a  gas,  with  absolution  of  heat 
at  constant  temperature.  The  constant  temperature  may  be  called 
the  transition  temperature ;  the  heat  absorbed  is  called  the  latent 
lieat  of  the  transition.  The  latter  name  is  due  to  Joseph  Black, 
the  discoverer  of  the  phenomenon  (1757);  he  appears  to  have 
regarded  the  heat  as  existing  "  latent "  in  the  body  in  some  sort 
of  chemical  combination,  just  as  "fixed  air"  exists  latent  in 
chalk.  In  both  cases  the  entity  has  lost  its  properties  by  chemical 
combination,  but  may  be  set  free  again  in  a  suitable  way. 

There  is  absorption  of  latent  heat  not  only  in  "  physical  " 
changes  of  state  (fusion,  evaporation),  but  also  in  many  chemical 
reactions  which  occur  at  a  transition  temperature.  In  all  cases 
the  transition  temperature  is  more  or  less  dependent  upon  the 


THERMOMETRY  AND  CALORIMETRY  19 

pressure,  and  the  latent  heat  depends  on  the  temperature  of 
transition. 

Definition. — The  heat  absorbed  in  producing  a  change  of 
physical  state  or  chemical  composition  of  a  system,  at  constant 
temperature  and  pressure,  is  called  the  latent  heat  of  the  given 
transition,  and  is  measured  by  the  number  of  calories  absorbed 
during  the  transition  of  unit  mass  of  the  substance  from  the 
initial  to  the  final  state. 

If  referred  to  one  gram  as  unit  mass  it  will  be  denoted  by 
L ;  if  to  a  molecular  weight  in  grams,  by  A  =  ML,  where 
M  =  molecular  weight. 

According  to  the  nature  of  the  transition,  we  have  different 
latent  heats: 

(1)  Latent  Heats  of  Fusion,  representing  the  heat  absorbed  in 
the  transition 

[Solid] >  [Liquid] 

at  a  standard  pressure,  usually  one  atmosphere. 

(2)  Latent  Heats  of  Evaporation,  absorbed  in  the  transition 

[Liquid] »  [Vapour], 

usually  measured  at  the  boiling-point  under  atmospheric 
pressure. 

(3)  Latent  Heats  of  Sublimation,  referring  to  the  change 

[Solid] >  [Vapour] ; 

e.g.,  camphor,  iodine,  etc.,  may  pass  directly  into  vapour  on 
heating. 

(4)  Latent  Heats  of  Dissociation,  absorbed  in  such  changes  as 
CaC03 »  CaO  +  C02  (Horstmann,  1869). 

(5)  Latent  Heats  of  Allot ropic  Change,  such  as 

Sa >S^at95c-6. 

(6)  Latent  Heats  of   Transition    (in  the  narrower  sense),  in 
which  a  system  of  substances  A  passes  over  into  another  system 
B   at  a  definite  temperature  and  pressure  with   absorption  of 
heat: 

Na^SOi-lOHaO >  Xa2S04+10H20  at  35°. 

Na2S04.10H20+  MgS04.7H20 >  NaMg(S04)2.4H20  +  13H20. 

CuCl2.2KC1.2H20 »CuCl2.KCl+KCl+2H20. 

In  all  the  examples  (1 — 6)  we  have  two  (or  more)  substances 

c  2     • 


20  THEKMODYNAMICS 

concerned  which  can  be  separated  from  each  other  mechanically, 
e.g.,  ice  and  water ;  CaC03,  CaO,  and  C02.  Such  homogeneous 
bodies  constituting  a  heterogeneous  complex  are  called  phases 
(Gibbs,  1876). 

The  transition  from  one  phase  to  another  is  called  phase- 
transition,-  it  takes  place  at  a  definite  temperature  (under  a 
specified  pressure)  with  the  absorption  of  an  amount  of  heat  A 
for  each  gram-molecule,  or  mol,  passing  over.  When  the  various 
phases  of  a  system  can  exist  side  by  side  for  an  indefinite  period 
they  >re  said  to  be  in  equilibrium. 


CHAPTER  II 

THE    FIRST    LAW    OF    THERMODYNAMICS     AND    SOME    APPLICATIONS 

12.     Force  and  Work. 

The  science  of  Mechanics  is  concerned  with  the  strict  defini- 
tion of  force  and  the  consequences  of  this  definition. 

In  daily  life  we  give  the  name  "  force  "  to  the  muscular  effort 
called  forth  in  supporting  a  weight,  or  in  maintaining  a  spring  or 
elastic  cord  in  a  condition  of  strain.  Each  of  these  systems 
exerts  a  force  in  opposition  to  the  muscular  effort,  and  the  pair 
of  forces  is  said  to  constitute  an  action  and  reaction  or  a  stress. 
The  legitimacy  of  this  nomenclature  is  assured  by  the  fact  that 
the  strained  spring  is  able  to  support  the  weight.  From  this 
point  of  view  the  weight  of  a  body  is  a  force.  If  we  regard  a 
given  extension  of  a  spiral  spring  as  brought  about  by  a  definite 
force,  it  is  found  that  the  weight  of  a  "given  body  is  not  an 
absolute  constant,  but  varies  slightly  with  the  latitude  and  the 
altitude  above  sea-level.  For  the  purpose  of  comparison,  all 
weights  are  reduced  to  latitude  45°  and  sea-level.  It  is  also 
found  that  the  changes  of  weight  are  in  the  same  ratio  for  the 
unit  of  weight  and  for  all  other  bodies,  so  that  if  the  weights  of 
two  bodies  are  equal  in  one  position  they  remain  equal  in  all 
olher  positions. 

An  unsupported  body  at  once  falls  towards  the  earth,  and  its 
velocity  increases  continuously  as  it  descends.  The  increase  of 
velocity  in  unit  time  was  found  by  Galilei  (1638)  to  be  constant ; 
it  is  called  the  acceleration  of  gravity  (</). 

In  the  C.G.S.  system  of  units  : 

y  =  980*53  —'2, 
sec.2 

at  latitude  45°  and  sea-level.  The  acceleration  of  gravity  varies 
slightly  from  place  to  place,  but  it  has  been  shown  experimentally 
that  it  varies  at  the  same  rate  as  the  weight  of  the  body,  so  that 


22  THERMODYNAMICS 

the  ratio  of  the  weight  to  the  acceleration  of  gravity  is  constant. 
This  ratio  is  called  the  mass  of  the  body. 

The  generalisation  of  this  result  is  contained  in  the  funda- 
mental law  of  mechanics,  due  to  Newton  (1687),  which  states  that 
a  force  P  which  imparts  a  velocity  u  to  a  mass  m  in  the  time  t  is 
directly  proportional  to  the  mass  and  velocity,  and  inversely  pro- 
portional to  the  time,  or 

P  =  k.  m  ft]  =  k.  mf    .  .    (1) 


where  /is  the  acceleration,  and  k  is  a  constant  depending  only  on 
the  choice  of  the  fundamental  units. 

If  the  latter  are  fixed,  the  unit  of  force  is  denned  when  a  parti- 
cular value  is  given  to  k.  If  the  gram,  centimetre,  and  second 
are  adopted,  and  k  is  put  equal  to  1,  then  P  =  1,  so  that  the  unit 
of  force  is  that  force  which,  acting  on  a  mass  of  one  gram  for 
one  second,  imparts  to  it  a  velocity  of  one  centimetre  per  second. 
This  unit  was  called  by  Clausius  a  dyne. 

The  dimensions  of  force  are      -'L-  J  or     mstr'2     . 

If  the  force  is  variable,  the  time  of  its  action  must  be  taken 
as  infinitely  small  : 

du  d?s 


This  method  of  fixing  the  derived  unit  by  making  the  arbitrary 
constant  equal  to  unity  was  largely  extended  by  Gauss  and 
Weber. 

The  application  of  force  to  a  body  produces  in  general  two 
effects  :  in  the  first  place  it  gives  rise  to  an  alteration  of  the  posi- 
tion of  the  body  with  respect  to  other  bodies,  or  of  the  parts  of 
the  body  with  respect  to  each  other,  and  in  the  second  place  it 
sets  the  body  in  motion.  These  are  called  the  change  of  configura- 
tion, and  the  change  of  motion,  of  the  body,  the  latter  referring  to 
a  change  of  velocity. 

From  the  aspect  of  the  applied  force,  the  effect  has  been  the 
movement  of  its  point  of  application  ;  from  the  aspect  of  the 
body  there  is  a  change  of  its  position  and  velocity. 

Definition.  —  If  a  force  moves  its  point  of  application  in  the 
direction  of  the  force  it  is  said  to  do  work.  The  work  done  is 


THE   FIRST   LAW  OF   THERMODYNAMICS          23 

measured  by  the  product  of  the  force  and  the  distance  through 
which  the  point  of  application  moves  in  the  direction  of  the 
force. 

The  unit  of  work  is  the  work  done  by  a  uniform  force  of  one 
dyne  in  moving  its  point  of  application  over  a  distance  of 
one  centimetre ;  it  was  called  by  Clausius  an  ery.  The  dimensions 

of  work  are       '-^J-      or      ?»s2  t~-    I  . 

If  s  =  distance  traversed  under  the  action  of  the  force  P,  the 
work  done  is  : 

A  =  Ps,  if  the  force  is  uniform ; 

or  A  —  Pds,  if  the  force  is  variable. 
Hence,  generally, 

A  =  I Pds  =  \m  -y-^  ds. 

If  the  point  of  application  moves  in  a  direction  opposite  to 
the  direction  of  the  force,  the  work  done  by  that  force  is  nega- 
tive, or  work  is  done  on  the  force.  Pds  will  in  general  be  a 

line  integral,  since  the  directions  of  the  force  and  path  may 
differ.  The  element  of  work  will  then  be  P.  cos  a.  ds,  where  a 
is  the  angle  between  the  direction  of  motion  and  the  line  of 
action  of  the  force. 

13.     Energy  of  Mechanical  Systems. 

As  examples  of  work  done  in  changing  the  configuration  of  a 
system  we  may  take  that  spent  in  raising  weights,  stretching  or 
bending  springs,  extending  elastic  cords,  or  separating  magnets 
or  electrically  charged  bodies  under  the  influence  of  attractive 
forces.  In  all  cases  the  system  itself  has  acquired  a  store  of 
something  which  can  be  reconverted  into  work  by  allowing  the 
parts  of  the  system  to  yield  to  the  action  of  the  forces  operative 
between  them.  Thus  the  raised  weight  may  do  work  (e.g.,  raise 
another  wreight)  by  descending.  The  greatest  amount  of  work 
which  the  system  can  do  in  passing  from  its  actual  configuration 
to  any  other  configuration  adopted  as  a  standard  is  called  the 
Potential  Energy  of  the  system  with  reference  to  the  standard 
configuration.  Thus,  if  a  weight  of  P  dynes  be  raised  a  distance 


24  THERMODYNAMICS 

of  s  cm.  above  the  standard  configuration  on  the  earth's  surface, 
the  system  composed  of  the  weight  and  the  earth  has  acquired 
Ps  ergs  of  potential  energy,  since  Ps  ergs  of  work  could  be  done 
by  allowing  the  weight  to  descend  to  the  standard  configuration, 
say  by  raising  an  equal  weight  through  an  equal  distance  by 
means  of  a  cord  passing  over  a  frictionless  pulley. 

The  second  general  effect  on  a  body  which  has  work  done  upon 
it  by  forces  is  a  change  in  the  velocity  of  the  body  ;  if  the  latter 
is  initially  at  rest  it  may  be  set  in  motion,  if  already  in  motion 
its  velocity  may  be  increased.  Such  a  moving  body  may,  in 
being  brought  to  rest  relative  to  some  standard  body,  produce  a 
change  of  configuration  of  some  external  system,  "i.e.,  do  work. 
The  penetration  of  an  obstacle  by  a  bullet  and  the  process  of  pile- 
driving  are  examples  of  this  type  of  action.  We  shall  now 
calculate  the  work  done  by  a  force  in  increasing  the  velocity  of  a 
particle,  initially  moving  with  uniform  velocity  «o-  Let  the  force 
P  be  applied  for  a  time  t,  during  which  the  particle  moves  over  a 
distance  s  in  the  direction  of  the  force.  Let  the  final  velocity  be 
MI.  Then  : 

(force)  =  (mass)  X  (acceleration) 


Multiply  both  sides  by  the  identity  ds  =  -j-  dt  and  integrate  : 
d*s    ds 


Ps  =  $wwia  —  £?n«o2 

so  that  the  work  done  is  equal  to  the  increase  of  the  magnitude 
(^  mu?),  which  is  called  the  Kinetic  Energy  of  the  particle. 

If  the  particle  is  initially  at  rest,  «0  =  0  .  •  .  Ps  =  £  -wm2. 

The  dimensions  of  kinetic  energy  are  [?ns2£~2],  i.e.,  the  same  as 
those  of  potential  energy  or  mechanical  work.  The  dimensions 
of  the  magnitudes  appearing  on  the  right  and  left  are  there- 
fore identical,  an  agreement  which  must  always  characterise  a 
physically  accurate  equation. 

The  sum  of  the  kinetic  and  potential  energies  of  the  particle  is 
therefore  constant. 

In  passing  from  the  consideration  of  a  simple  massive  particle 
to  finite  masses,  which  may  exert  mutual  actions,  or  in  which 


THE   FIRST  LAW  OF   THERMODYNAMICS          25 

there  may  be  internal  forces  of  elasticity,  the  treatises  on  dyna- 
mics show  that  in  every  case  where  the  force  equations  are  valid, 
i.e.,  when  there  are  no  frictional  resistances,  the  kinetic  and 
potential  energies,  although  freely  interconvertible,  remain  con- 
stant in  combined  amount. 

In  actual  machines,  where  friction  is  present,  they  are  known 
to  diminish  steadily  during  action. 

Thus,  a  simple  pendulum  affords  an  example  of  an  ideal 
machine.  When  the  bob  is  in  its  highest  position  all  the  energy 
is  potential ;  when  in  its  lowest  position  all  the  energy  is 
kinetic.  In  passing  from  one  limiting  position  to  the  other  the 
energies  suffer  mutual  interconversion,  but  their  sum  is  always 
constant.  In  actual  pendulums,  the  sum  gradually  diminishes 
to  zero,  owing  to  friction.  We  now  inquire  into  the  fate  of  the 
lost  energy. 

14.     The  Forms  of  Energy. 

It  is  a  fact  of  experience  that  when  work  is  spent,  something 
else  appears  in  its  place,  and  that  when  work  is  produced,  some 
other  thing  disappears.  By  way  of  definition  we  shall  call  work, 
and  anything  obtainable  from  or  convertible  into  work,  forms  of 
energy. 

The  potential  and  kinetic  energies  of  mechanics  are  two  forms, 
since  they  satisfy  the  definition. 

Other  forms  of  energy  are  recognised  in  the  same  way. 

When  work  is  done  by  a  force  against  friction,  as  in  dragging 
a  weight  up  a  rough  incline,  or  projecting  a  mass  on  a  rough 
plane,  the  gam  of  potential  or  kinetic  energy  is  always  less  than 
the  work  done  by  the  force.  In  addition,  however,  a  rise  of 
temperature  is  observed  in  the  system,  or  in  those  parts  where 
the  friction  is  located — in  other  words,  Heat  is  produced.  This, 
being  obtained  from  work  spent  on  the  system,  is  a  form  of 
energy.  The  reverse  process,  in  which  heat  is  converted  into 
work,  is  utilised  in  all  steam  engines.  The  heating  of  a  rapidly 
moving  bullet  on  striking  a  target  is  an  instance  of  the  conversion 
of  kinetic  energy  into  heat.  The  reverse  process  occurs  in  the 
Trevelyan  rocker. 

The  ultimate  source  of  all  but  a  very  small  fraction  of  the 
energy  available  for  terrestrial  purposes  has  reached  the  earth 
from  the  sun  in  the  form  of  Radiant  Energy,  which  is  called  light 


26  THERMODYNAMICS 

if  its  quality  is  such  that  it  is  perceived  by  the  eye,  and  either 
radiant  heat  or  ultra-violet  radiation  if  this  is  not  the  case. 
Radiant  energy  is  transformed  into  heat  when  it  is  absorbed  by 
a  blackened  strip  of  platinum  on  which  it  is  incident,  and  since 
the  amount  of  heat  can  be  found  from  the  resulting  rise  of 
temperature,  this  apparatus — called  a  Bolometer — serves  to 
measure  the  amount  of  incident  radiation.  If  the  body  on  which 
radiation  is  incident  is  not  a  black  body,  a  portion  only  of  the 
radiation  is  converted  into  heat,  the  rest  is  either  transmitted  or 
reflected  unchanged,  or  changed  in  quality  ("  fluorescence "'), 
or  is  spent  in  producing  chemical  changes,  as  in  the  assimilative 
processes  of  green  plants,  the  processes  of  photograph}7,  bleaching 
by  exposure  to  light,  etc.  Another  type  of  radiant  energy  is  the 
electromagnetic  radiation  utilised  in  wireless  telegraphy ;  this, 
however,  is  merely  another  quality  of  radiation  which  differs 
from  radiant  heat  and  light  only  in  the  same  respect  that  these 
differ  from  each  other. 

Nearly  all  chemical  changes  can  be  effected  in  such  a  way  that 
an  evolution  (or  absorption)  of  heat  occurs  along  with  the  change 
of  composition  of  the  system,  this  phenomenon  being  usually 
regarded  as  indicative  of  chemical  change.  The  initial  material 
system,  such  as  the  wax  of  a  candle  and  the  oxygen  of  the  air, 
contains  more  (or  less)  energy  than  the  final  system,  and  this 
store  of  energy,  part  of  which  is  set  free  during  chemical  change 
into  the  final  system,  such  as  water  and  carbon  dioxide,  and 
usually  escapes  in  the  form  of  heat,  is  called  Chemical  Enerf/y. 

If  we  do  not  wish  to  include  radioactive  changes  under  this 
heading,  another  form  of  energy,  which  is  emitted  during  such 
changes  as  heat,  kinetic  energy,  and  electromagnetic  radiation, 
must  be  described.  All  substances  exhibiting  this  property  must 
contain  a  store  of  energy  which  might  be  called  Radioactive 
Energy.  This  emission  of  energy  apparently  has  its  origin  in 
deep-seated  changes  in  the  very  atoms  themselves,  which  differ 
from  chemical  changes  in  being  independent  of  temperature.  The 
atoms  of  other  elements  differ  only  from  those  of  radioactive 
elements  in  being  more  stable,  and  this  energy  may  therefore  be 
called  A tomic  Energy;  radioactive  elements  are  those  which  are 
capable  of  yielding  up  their  atomic  energy  in  other  forms. 

The  electric  current  supplied  by  power  companies  may  be 
utilised  in  producing  mechanical  work  (through  the  agency  of 


THE   FIRST  LAW  OF   THERMODYNAMICS          27 

electromotors),  in  heating  or  lighting,  and  in  producing  chemical 
effects  by  electrolysis.  It  has  therefore  associated  with  it  a  form 
of  energy,  and  this  is  called  Electrical  Energy. 

Iron  and  steel  have  the  capacity  of  taking  up  permanently  a 
peculiar  state,  in  which  they  show  polar  properties  and  are  said 
to  be  magnetised.  In  the  formation  of  magnets,  energy — either 
in  the  form  of  mechanical  work,  as  in  magnetisation  by  contact 
with  other  magnets,  or  of  electrical  energy  in  the  formation  of 
electromagnets — is  supplied  from  outside,  and  we  may  say  that 
this  energy  is  transformed  into  Magnetic  Energy  in  the  magnetised 
system. 

If  oil  and  water  are  agitated  together,  the  former  is  broken  up 
into  globules  which  are  disseminated  through  the  water.  If  the 
mixture  is  now  allowed  to  stand,  the  globules  unite  into  larger 
drops,  and  finally  the  latter  coalesce  into  a  homogeneous  layer. 
During  the  latter  changes,  the  masses  of  oil  are  observed  to  be,  iu 
many  cases,  violently  agitated,  and  thus  kinetic  energy  is  pro- 
duced'. When,  therefore,  a  surface  of  separation  between  two  or 
more  bodies  is  increased  or  diminished,  a  definite  amount  of 
energy  must  be  given  to,  or  may  be  abstracted  from,  the  system. 
The  energy  a  system  possesses  in  virtue  of  its  surfaces  of  separa- 
tion is  called  Surface  Energy.  Since  work  must  be  spent  in 
stretching  a  thin  film  of  soap  solution,  for  example,  it  is  evident  that 
this  energy  must  be  located  "  hi  "  the  surface  of  separation  itself. 

Thus,  bodies  or  systems  may  possess  energy  because  they  are 
in  a  certain  state  of  configuration,  electrification,  or  magnetisa- 
tion, or  because  they  are  in  motion,  or  are  hot,  or  luminous, 
or,  finally,  because  they  are  capable  of  undergoing  chemical 
or  radioactive  change. 

We  have  therefore  considered  the  following  forms  in  which 
energy  is  recognised  : 

(1)  Kinetic  energy  of  masses  in  motion. 

(2)  Potential  energy  of  systems  of  forces. 

(3)  Heat  energy. 

(4)  Radiant  energy. 

(5)  Chemical  energy. 

(6)  Atomic,  or  Radioactive,  energy. 

(7)  Electrical  energy. 

(8)  Magnetic  energy. 

(9)  Surface  energy. 


28  THERMODYNAMICS 

The  question  as  to  whether  there  are  other  forms  of  energy  in 
addition  to  these  can  only  be  answered  by  experiment.  At 
present  no  evidence  has  been  adduced  for  their  existence, 
although  speculation  has  not  been  lacking  (cf.,  for  example, 
W.  Ostwald,  Die  Energie,  Leipzig,  A.  Earth,  1908). 

15.     The  Mechanical  Equivalent  of  Heat. 

There  is  a  fixed  relation  between  the  measure  of  a  quantity  of 
work  and  that  of  the  quantity  of  heat  obtained  from  it  by 
complete  conversion.  If  these  two  measures  are  expressed  in 
terms  of  the  erg  and  the  calorie  respectively  as  units,  there  will 
also  be  a  relation  between  the  erg  and  the  calorie.  Heat,  con- 
sidered as  a  form  of  energy,  may  be  measured  in  ergs,  i.e.,  in 
work  units,  and  to  convert  the  measure  of  a  quantity  of  heat 
expressed  in  calories  into  the  measure  of  the  same  quantity 
expressed  in  ergs,  we  must  find  the  number  of  times  the  erg  is 
contained  in  the  calorie,  and  multiply  this  by  the  measure  of  the 
given  quantity  of  heat  in  calories.  It  is  a  relation  between  units 
which  is  involved. 

Definition. — The  Mechanical  Equivalent  of  Heat  (J)  is  the 
number  of  ergs  of  work  which,  if  completely  converted  into  heat, 
would  give  rise  to  one  calorie. 

That  such  a  mechanical  equivalent  exists  is  a  consequence  of 
the  fact  that  heat  and  work  are  interconvertible.  The  further 
fact  that  it  is  a  fixed  constant  and  independent  of  the  process  of 
conversion  was  proved  by  the  experiments  of  Joule  (1843-1880), 
referred  to  below.  In  the  determination  of  this  constant  two 
measurements  are  required  : 

(1)  The  number  of  ergs  of  work  which  disappear  in  a  given 
change  and  are   completely  converted  into  heat. 

(2)  The  number  of  calories  of  heat  produced. 

J.  R.  Mayer  (1842)  made  the  first  calculation  of  the  mechanical 
equivalent  of  heat  by  comparing  the  work  done  on  expansion  of 
air  with  the  heat  absorbed. 

(1)  Joule's  Researches  (1843-1880).— (a)  The  heat  produced  in 
a  coil  of  wire  by  induction  currents  set  up  on  rotation  between  the 
poles  of  an  electromagnet  was  communicated  to  water  in  which 
the  coil  was  placed,  and  was  compared  with  the  work  done  by 
falling  weights  in  rotating  the  coil  (1843). 


THE   FIRST  LAW  OF   THERMODYNAMICS  29 

(b)  The  heat  produced  by  stirring  water,  oil,  and  mercury  was 
compared  with  the  work  done  by  falling  weights  which  actuated 
the  stirrer  (1845-9). 

(c)  The  work  done  in  compressing  air  was  compared  with  the 
heat  produced  (1844-5). 

(d)  The  heat  produced  by  the  passage  of  an  electric  current 
through  a  coil  of  wire  immersed  in  a  calorimeter  was  compared 
with  the  energy  spent  by  the  current  (1867). 

The  mean  of  Joule's  results,  recalculated  in  terms  of  modern 
temperature  standards  and  electrical  units,  gave  : 

J  =  4-1714  X  107  ergs  per  15°  cal.  from  (a)— (c). 
J  =  4-1665  X  107  ergs  per  15°  cal.  from  (d> 

(2)  Later  Work.— (a)  The  water-stirring  experiments  have 
been  repeated,  and  the  results  show  that  Joule's  number  is 
certainly  about  0'3  per  cent,  too  low : 

H.  Rowland  (1879-80)  ....  4*188  X  107 
C.  Miculescu  (1892)  ....  4181  X  107 
0.  Reynolds  and  W.  H.  Moorby  (1898)  .  4'1845  x  107 

(b)  The  electrical-heating  method  is  apparently  the  most 
accurate,  and  has  been  favoured  by  later  workers  : 

E.  H.  Griffiths  (1893)    ....  4'1856  X  107 

A.  Schuster  and  W.  Gannon  (1895)       .  41859  X  107 

H.  Callendar  and  H.  T.  Barnes  (1902)  .  4 '1851  X  107 

C.  Dieterici  (1905)         ....  41938  X  107 

Dieterici's  number  is  much  higher  than  the  others,  which  are  in  good 
agreement;  this  experimenter  used  the  Bunsen  ice  calorimeter,  which  is  a 
very  uncertain  instrument.  His  results  are,  however,  included  in  the  mean 
adopted  below. 

Rowland  first  observed  that  the  value  of  J  varies  with  the 
temperature,  and  if  the  mechanical  equivalent  of  the  15°  calorie 
be  taken  as  standard,  the  ratio  of  the  value  of  J  at  any  tempera- 
ture to  this  gives  the  specific  heat  of  water  at  that  temperature. 
He  found  the  curious  result  that  the  specific  heat  of  water  had  a 
maximum  value  at  about  30°  ;  Callendar  and  Barnes  located  this 
at  37G'5. 


30 


THERMODYNAMICS 


We  shall  take : 

J  =  4*188  (+  -002)  X  107  ergs  per  15°  cal. 

which  is  the  value  computed  by  Luther  and  Scheel  (Zeitschr. 
Elektrochem.,  1903,  9,  686).  On  account  of  the  inclusion  of 
Dieterici's  value,  this  number  is  probably  slightly  too  great. 

Table  of  the  amounts  of  heat,  required  to  warm  1  gram  of  water 
from  (9  —  |)°  to  (0  +  £)°  on  the  hydrogen  scale. 


e° 

15°  cal. 

Erg  x  107. 

0 

1-010 

4-229 

5 

1-004 

4-205 

10 

1-002 

4-196 

15 

i-ooo 

4-188 

20 

0-999 

4-184 

25 

0-998 

4-180 

30 

0-997 

4-176 

35 

0-997 

4-175 

40 

0-997 

4-175 

45 

0-997 

4-175 

50 

0-998 

4-180 

55 

0-999 

4-182 

60 

0-999 

4-184 

65 

0-999 

4-186 

70 

1-000 

4-188 

75 

1-001 

4-190 

80 

1-001 

4-192 

85 

1-002 

4-196 

90 

1-003 

4-200 

95 

1-004 

4-203 

100 

1-004 

4-205 

1  mean  calorie  = 

1-0002  15°  cal. 

4-1886  erg.  X  107 

The  entire  agreement  between  the  values  of  the  mechanical 
equivalent  of  heat  obtained  by  many  different  methods  establishes 
the  proposition  that  it  is  independent  of  the  process  in  which  the 
conversion  of  work  into  heat  occurs,  and  depends  solely  on  the 
choice  of  the  units  of  these  two  magnitudes.  This  result  was 
first  established  by  Joule. 


THE   FIRST   LAW  OF    THERMODYNAMICS  31 

It  has  recently  been  proposed  to  specify  quantities  of  heat  directly  in  work 
units  in  terms  of  the  joule  (j.)  =  107  erg,  and  the  kilojoule  (kj.)  =  1010  erg 
(Ostwald ;  T.  W.  Eichards).  Whilst  this  has  an  advantage  from  the  point 
of  view  of  uniformity  and  simplicity,  it  suffers  from  the  disadvantage  that 
the  whole  mass  of  experimental  data  must  be  recalculated  every  time  a 
more  exact  value  of  J  is  determined. 

With  the  present  value  1  cal.  =  4-188  j.,  1  kj.  =  238'S  cal. 

16.  The  First  Law  of  Thermodynamics. 

The  following  statement  is  a  consequence  of  the  experimental 
results  of  Joule,  that  is,  of  the  existence  of  a  unique  mechanical 
equivalent  of  heat,  and  is  known  as  the  First  Law  of  Thermo- 
dynamics : 

When  equal  quantities  of  work  are  produced  by  any  means 
from  purely  thermal  sources,  or  spent  in  purely  thermal 
effects,  equal  quantities  of  heat  are  put  out  of  existence,  or  are 
generated. 

If  Q  units  of  heat  appear  (or  disappear)  in  any  process,  and 
A  units  of  wrork  disappear  (or  appear)  simultaneously,  then, 
provided  no  other  forms  of  energy  appear  or  disappear,  i.e.,  if  the 
work  is  produced  or  spent  in  "  purely  thermal "  processes,  the 
first  law  of  thermodynamics  asserts  that : 

A  =  JQ         .         .         .         .         (1), 

where  J  is  a  constant  depending  only  on  the  choice  of  the  units 
in  which  A  and  Q  are  expressed. 

It  is  particularly  to  be  emphasised  that  equation  (1)  holds  good 
only  when  the  conversion  of  heat  into  work,  or  vice  versa,  occurs 
directly,  i.e.,  when  no  other  forms  of  energy  are  appearing  or 
disappearing  at  the  same  time.  No  conclusion  as  to  the  ratio  of 
transformation  of  two  forms  of  energy  can  be  drawn  unless  the 
transformation  occurs  directly ;  a  process  of  the  latter  type  we 
shall  designate  an  aschistic  process,  because  the  energy  change 
occurs  along  an  unbranched  path. 

17.  Cyclic  Processes. 

If,  after  any  process,  or  series  of  processes,  a  system  returns  to 
its  initial  state,  it  is  said  to  have  undergone  a  cycle  of  changes, 
and  the  process  is  called  a  cyclic  process  (S.  Carnot,  1824). 

Thus,  if  a  mass  of  air  in  a  cylinder  is  compressed  or  expanded, 
with  or  without  simultaneous  heating  or  cooling,  in  any  way 


32  THERMODYNAMICS 

whatever,  but  is  finally  restored  to  its  initial  volume  and 
temperature,  a  cycle  of  changes,  or  a  cyclic  process,  has  been 
executed. 

In  any  cyclic  process,  let  a  quantity  of  heat  Q  be  absorbed,  and 
a  quantity  of  work  A  be  done,  by  the  system.  Heat  emitted,  or 
work  done  on  the  system,  is  to  be  reckoned  with  a  negative  sign. 
Then  the  heat  absorbed  is  equivalent  to  the  work  done,  in  the 
sense  explained : 

A  =  JQ, 

i.e.,  such  a  cyclic  process  is  an  aschistic  process.  If  this  were  not 
the  case,  it  would  be  possible  to  produce  or  destroy  unlimited 
quantities  of  heat  or  work  without  giving  rise  to  any  other 
changes  whatever,  which  is  contrary  to  the  First  Law  of 
Thermodynamics. 

18.     Schistic  Processes  :  Intrinsic  Energy. 

A  system  is  said  to  be  in  equilibrium  in  a  thermodynamic 
sense  when  the  position  of  each  of  its  parts  relative  to  a  fixed 
point,  and  the  state  of  that  part,  remain  unchanged  with  lapse  of 
time. 

The  first  condition  asserts  that  the  kinetic  energy  of  the 
system,  relative  to  the  fixed  point,  is  constantly  zero,  and  refers 
to  mechanical  equilibrium. 

By  the  term  "  state  "  we  refer  to  such  data  as  the  chemical 
composition  of  the  parts  of  the  system  (including  allotropy  and 
isomerism),  their  state  of  electrification,  magnetisation,  stress  or 
strain,  their  state  of  division,  temperature,  and  the  like,  and  the 
second  condition  generalises  the  statement  of  equilibrium. 

If  a  system  is  not  in  equilibrium  its  state  changes  with  lapse 
of  time.  A  system  in  equilibrium  may  also  be  caused  to  change 
by  means  of  external  actions. 

We  now  suppose  that  a  system  undergoes  any  change,  and 
that  no  energy  changes  occur  outside  the  system  except  the 
disappearance  of  a  quantity  of  heat  Q  and  the  performance  of  a 
quantity  of  work  A.  If  the  final  state  is  the  same  as  the  initial 
state,  the  process  must,  by  definition,  have  been  a  cyclic  process, 
and  hence 

A  =  JQ         .         .         .         .         (a). 

If  the  final  state  is  different  from  the  initial  state,  no  relation 


THE   FIKST   LAW   OF   THERMODYNAMICS          33 

between  A  and  Q  can  be  ascertained  a  priori.  We  shall  suppose,  in 
this  case,  that  the  magnitudes  A  and  Q  have  been  experimentally 
determined ;  two  possibilities  are  now  open  : 

either  (i.)  A  =  JQ         .         .         .         .        (a'), 

i.e.,  the  process  is  aschistic  but  not  cyclic ; 

or  (ii.)  A  is  not  equal  to  JQ         .         .         .         (ft), 

i.e.,  the  process  is  not  aschistic. 

Since  the  conditions  exclude  the  possibility  of  any  other  energy 
changes  occurring  outside  the  system,  we  conclude  that  there  has 
been  a  change  in  some  amount  of  energy  which  is  located  in  the 
system  itself.  We  must  therefore  regard  the  system  as  possessing 
a  store  of  energy  which  may  be  increased  or  diminished  by 
changes  in  the  physical  or  chemical  state  of  the  system,  and  we 
shall  call  this  the  intrinsic  energy  of  the  system,  since,  as  we  shall 
prove  in  a  moment,  it  depends  solely  on  the  state  of  the  particular 
system  in  which  it  is  located.  The  change  of  intrinsic  energy  we 
shall  now  define  as  follows  : 

Definition  of  Intrinsic  Energy. — Let  there  be  given  any  system 
of  bodies,  and  let  the  system  undergo  any  change  whatever,  so 
that  it  passes  from  a  given  initial  state  [1]  to  a  final  state  [2],  the 
only  condition  imposed  on  the  states  [1]  and  [2]  being  that  they 
shall  be  consistent  with  the  physical  properties  of  the  system. 

In  this  change  there  will  be  in  general  a  finite  amount  of  heat 
withdrawn  by  the  system  from  its  environment,  which  may  be 
absorbed  in  various  parts.  The  total  absorbed  heat  we  shall 
denote  by  2Q.  At  the  same  time  the  parts  of  the  system,  exert- 
ing forces  on  external  bodies,  may  perform  mechanical  work. 
The  total  mechanical  work  done  by  the  system  on  external  bodies 
we  shall  denote  by  2A. 

Thus  in  the  evaporation  of  1  gram  of  water  under  a  piston,  the  heat 
2251  x  107  erg  is  absorbed  in  "raising  steam,"  and  the  work  1662  x  107 
erg  is  done  by  the  piston  against  the  atmospheric  pressure.  Thus 
J2Q  -  2A  =  (2251  -  1662)  X  10?  =  589  X  107  erg. 

Then  if  Ui  be  the  intrinsic  energy  of  the  system  in  its  initial 
state,  and  U2  the  intrinsic  energy  of  the  system  in  its  final  state, 
we  define  the  increase  of  intrinsic  energy  of  the  system  by  the 
equation  : 

Ua  -  Ui  =  J2Q  -  2A         .         .         .         .     (c) 


84  THERMODYNAMICS 

If  Q  is  measured  in  the  same  units  as  A  (e.y.,  in  ergj),  we 
may  omit  the  factor  J,  provided  we  remember  always  that 
quantities  of  heat  are  to  be  expressed  in  work  units  in  the  equa- 
tions which  follow.  Then  : 

U2  -  LT!  =  2Q  -  2A          .        .        .         .     (c') 

(U2  —  Ui)  is  a  definite  magnitude,  since  both  2Q  and  2A  are 
experimentally  determinable. 

If  (SQ  —  2A^<  0,  then   (lJ2  -  Ui)<  0,  i.e.,  the  intrinsic 

energy  has  increased,  diminished,  or  remained  constant,  accord- 
ing as  the  absorbed  heat  is  greater  than,  less  than,  or  equal  to, 
the  external  work  done. 

An  aschistic  process  implies  constancy  of  intrinsic  energy. 

The  actual  state,  and  absolute  amount,  of  intrinsic  energy 
existing  in  a  body,  or  system  of  bodies,  are  things  which  lie  quite 
outside  the  range  of  pure  thermodynamics.  This  indefmiteness 
has,  however,  not  the  slightest  influence  on  the  stringency  of  the 
definition,  since  we  can  proceed  as  in  the  definition  of  electrostatic 
potential,  and  choose  any  convenient  standard  state  of  the  body, 
and  use  the  term  "intrinsic  energy"  with  reference  to  this 
standard  state. 

If  U0  is  the  absolute  amount  of  intrinsic  energy  contained  in  a 
system  (with  reference  to  a  state  of  absolute  zero  of  energy)  in 
an  arbitrary  standard  state,  and  if  in  any  change  from  a  state 
[1]  to  a  state  [2]  the  total  amounts  of  heat  absorbed  and  work 
done  are  2Q  and  SA  respectively,  we  have  : 

2Q-2A  =  [Ux  +  Uoi;  =  U2  -  Ux, 

where  ~UX  is  the  intrinsic  energy  in  any  intermediate  state  x,  with 
reference  to  the  standard  state  (Lord  Kelvin,  1851). 

It  will  be  observed  that  the  definition  of  intrinsic  energy  by 
means  of  the  equation  (c)  implies  in  itself  no  physical  law,  since 
the  value  of  (U2— Ui)  can  always  be  chosen  so  as  to  make  the 
values  of  2Q  and  2A  satisfy  the  equation.  We  shall  now  show 
that  the  value  of  (Ua  —  Ui)  is  uniquely  so  defined,  and  is  quite 
independent  of  the  way  in  which  the  process  is  executed.  This  is 
a  physical  law,  which  we  shall  call  the  Principle  of  Conservation 
of  Energy. 

It  may  be  formally  expressed  as  follows. 


THE  FIRST  LAW  OF   THERMODYNAMICS          35 

19.     The  Principle  of  Conservation  of  Energy. 

The  change  of  intrinsic  energy  of  a  system  undergoing  any  change 
of  state  depends  solely  on  the  initial  and  final  states  of  the  system, 
and  is  independent  of  the  manner  in  which  the  change  from  the  one 
state  to  the  other  is  effected. 

Proof. — Let  the  change  from  state  [1]  to  state  [2]  be  effected  in 
different  ways,  which  are  denoted  by  (a),  (/J)  .  .  .,  (A);  if  the 
changes  of  intrinsic  energy  are  not  equal,  let  them  be  AUa, 
AUfl  .  .  . ,  AUA,  for  the  different  paths.  Let  AUa,  AU/s  beanyivfo 
of  these  values.  We  shall  now  assume  that  at  least  one  process  is 
possible  whereby  a  reverse  change  from  the  state  [2]  to  the  state 
[1]  may  be  effected;  let  this  process  be  denoted  by  ((/>).  Now 
perform  the  following  cyclic  process  : 

(i.)  Pass  from  [1]  to  [2]  along  (a),  and  return  along  (<£).  The 
diminution  of  intrinsic  energy  =  AU<^  —  AUtt. 

(ii.)  Pass  from  [1]  to  [2]  along  (/?,)  and  return  along  (<£).  The 
diminution  of  intrinsic  energy  is  AU^  —  AUg. 

The  system  is  now  in  its  initial  state,  and  since  no  energy  has 
been  obtained  from,  or  given  to,  other  systems,  except  that 
derived  from  the  given  system,  we  see  that  unless  the  two  quan- 
tities (AU^  —  AU0),  (AU^,  —  AU0)  are  both  zero,  a  quantity  of 
energy  will  have  been  gained  or  lost  without  any  other  change 
having  occurred. 

This  is  in  contradiction  to  the  First  Law  of  Thermodynamics, 

and  hence :          AU^,  —  AUa  =  AU^  —  AUp  =  0, 
or  AUa  =  AU0  =  AU*, 

and  since  (a),  (/3)  are  any  two  possible  paths,  the  proposition  is 
true  for  all  possible  paths. 

It  must  be  observed  that  the  formal  proof  of  the  theorem  depends  on  the 
possibility  of  returning  to  the  initial  state  along  at  least  one  path  such  as  ($). 
The  extension  of  the  theorem  to  vital  processes,  phosphorescence,  and  radio- 
active changes,  which  have  not  yet  been  reversed,  must  therefore  bs  regarded 
as  inductive,  although  highly  probable. 

If  the  given  change  is  infinitesimal,  we  shall  denote  2Q  and 
2  A  by  SQ  and  SA  respectively. 

Corollary  1.  For  any  two  states  of  a  system,  say  (1)  and  (2), 
the  value  of  the  integral : 

3-*A) 

D  2 


36  THERMODYNAMICS 

is  independent  of  the  manner  in  which  the  change  from  [1]  to 
[2]  is  effected,  i.e.,  (SQ  —  8A)  =  dU  is  a  perfect  differential 
(Lord  Kelvin,  1851). 

Corollary  2.  For  a  cyclic  process  : 

(J)  SA  =  (j>Q, 

where  SA,  SQ  denote  the  elements  of  work  done  and  heat 
absorbed  during  any  infinitesimal  part  of  the  cycle,  and  (  )  is 
taken  as  signifying  an  integration  extended  round  the  cycle. 

For  (J)8A  =  (J)SQ  -  (f)dU,  and  (J)dU  =  0,  by  reason  of 
the  identity  of  the  initial  and  final  states.  The  magnitudes 
(I)SA,  (|)SQ,  denoting  the  nett  work  done,  and  heat  absorbed,  in 
the  cyclic  process,  are  however,  not  usually  zero,  and  may  have 
very  different  values  (provided  (I )&Q  —  (|)8A  is  always  zero), 

according  to  the  particular  way  in  which  the  process  is  conducted. 
The  same  applies  to  the  magnitudes  2A  and  2Q ;  for  the  same  ini- 
tial and  final  states,  the  work  done  and  heat  absorbed  may  have  very 
different  values  according  to  the  way  in  which  the  change  is  effected ; 
these  pairs  of  values  must,  however,  always  satisfy  the  equation  : 

2Q  —  2A  =  constant,  for  fixed  terminal  states. 

This  very  important  theorem  was  recognised  by  R.  Clausius 
in  1850,  although  he  did  not  at  the  time  give  the  very  simple 
interpretation,  in  terms  of  the  conception  of  intrinsic  energy, 
which  was  brought  forward  by  Lord  Kelvin  a  year  later. 

8Q,  SA  are  what  are  called  imperfect  differentials ;  (8Q  —  &A) 
=  dU  is  a  perfect  differential  (cf.  H.  M.,  §§  57,  115).  As  a  matter 
of  fact,  &Q,  SA  are  not  differentials  of  functions  of  the  independent 
variables  of  state  at  all,  and  might  be  written  q,  a. 

The  path  of  change  is  usually  more  or  less  open  to  arbitrary 
choice,  and  we  shall  next  consider  one  or  two  important  cases,  in 
each  of  which  some  kind  of  constraint  is  put  upon  the  energy 
changes : 

(1)  The  change  is  aschistic: 

2A  =  2Q 
.-.  AU  =  0 

orU  =  constant. 


THE   FIRST  LAW   OF   THERMODYNAMICS          37 

A  particular  case  is  a  cyclic  process ;  an  example  of  a  non- 
cyclic  aschistic  change  is  afforded  by  the  expansion  of  an  ideal 
gas  at  constant  temperature  (§  71). 

(2)  The  change  is  adiabatic,  i.e.,  the  transfer  of  heat  to  or 
from  the  system  from  outside  is  prevented,  say  by  enclosing  the 
system  in  a  perfectly  non-conducting  envelope.     Then  : 

2Q  =  0 
.'.2A=  AU, 

so  that  the  external  work  is  performed  wholly  at  the  expense  of 
the  intrinsic  energy  of  the  system. 

(3)  The  change  is  adi/namic,  i.e.,  no  external  work  is  performed. 
Thus,  we  may  imagine  the  system  enclosed  in  a  perfectly  rigid 
envelope  which,   however,  permits  the  free  passage   of   heat. 
Then: 

2A=:  0 
.'.  AU  =  2Q, 

so  that  the  heat  absorbed  goes  wholly  to  increase  the  intrinsic 
energy.  This  provides  a  method  of  determining  the  latter 
magnitude,  since  2Q  is  experimentally  measurable. 

(4)  A  system  which  is  cut  off  from  communication  of  all  kinds 
of   external  energy  is  called  an  isolated  system.     A  system  of 
bodies  contained  in  a  vessel  with  perfectly  rigid  non-conducting 
walls  is  such  a  system.     In  this  case  2Q  =  2A  =  0  .  * .  V%  =  Ui 
.•.  U  =  constant.      The  Principle  of  Conservation  of  Energy  is 
usually  expressed  in  the  form  that  the  intrinsic  energy  of  an 
absolutely  isolated  system  of  bodies  is  constant  and  independent 
of  all  changes  of  state  which  may  occur  subject  to  the  condition 
that  the  system  remains  isolated.     Since  in  this  case  we  have 
absolutely  no  means  of  examining  the  energy  content  of   the 
system,  the  statement  appears  somewhat  indefinite. 

20.     Pressure. 

When  two  bodies  are  placed  in  contact  there  is  in  general  a 
distribution  of  force  over  the  area  of  contact.  Such  a  distribu- 
tion of  force  over  an  area  is  called  a  thrust,  and  if  the  force  at  all 
points  is  normal  to  the  area,  the  thrust  per  unit  area  is  called 
the  pressure.  If  the  force  is  inclined  at  an  angle  6  to  the  normal 
to  the  area,  the  resolved  part,  P  cos  0,  only  is  taken  into  account. 
Since  in  general  the  thrust  is  not  uniformly  distributed,  we  must 


38  THERMODYNAMICS 

introduce  the  conception  of  the  pressure  at  a  point  on  an  area. 
Let  P  be  the  force  acting  normally  to  the  centre  of  gravity  of  a 
very  small  plane  area  da.  Then  if  P/da  tends  to  a  finite  limit  as 
da  approaches  zero,  this  is  denned  as  the  pressure  at  the  point 
which  is  the  centre  of  gravity  of  da. 

We  may  also  speak  of  the  pressure  at  a  point  in  the  interior  of 
a  mass  of  liquid  or  gas,  because  if  a  very  small  plane  area  a-  is  drawn 
around  that  point  as  centre  of  gravity,  and  all  the  fluid  removed 
from  the  immediate  vicinity  of  one  side,  a  definite  force  P  must 
be  applied  to  keep  the  area  in  position.  From  the  principle  of 
reaction  we  see  that  each  of  the  two  portions  of  fluid  divided  by 
an  imaginary  plane  o-  exerts  a  pressure  P/o-  on  the  other.  Such 
a  pair  of  equal  and  opposite  forces  is  called  a  stress. 

Stresses  may  also  exist  in  the  interior  of  solid  bodies,  and  are  considered 
in  the  theory  of  elasticity. 

This  pressure  is  the  distribution  of  molecular  impacts  on  the 
surface.  If  da  is  smaller  than  a  finite  but  very  small  size  the 
pressure  loses  its  significance,  and  is  replaced  by  a  sporadic 
bombardment  by  individual  molecules  at  intervals  comparable 
with  the  time  in  the  mean  free  path.  This  occurs,  for  instance, 
in  the  spinthariscope  of  Crookes.  The  condition  that  P/V?a 
approaches  a  finite  limit,  independent  of  the  original  size  of  da, 
when  da  — >  0,  is  apparently  in  contradiction  to  the  physical  pro- 
perties of  the  system.  A  similar  difficulty  arises  in  the  definition 
of  the  density  at  a  point  in  a  medium  of  varying  density,  by 
considering  the  mass  of  a  volume  element  around  the  point. 
The  space  would  sometimes  be  occupied  by  a  small  number  of 
molecules,  at  other  times  by  a  larger  number,  and  occasionally 
by  none  at  all.  The  requirements  of  molecular  physics  and  of 
the  infinitesimal  calculus  are  therefore  apparently  in  direct 
opposition.  The  solution  of  the  difficulty  becomes  apparent 
as  soon  as  we  remember  how  exceedingly  minute  are  the 
individual  molecules  in  comparison  with  any  "finite  portion" 
of  a  body.  The  infinitesimal  element  may  be  chosen  so  small 
that  it  satisfies  the  conditions  imposed  by  the  mathematical 
analysis,  and  yet  remains  sufficiently  large  to  be  physically 
homogeneous.  Such  an  element  of  volume,  or  generally  such  an 
element  of  a  molecular  system,  may  be  called  a  physically  small 
element  (Leathern :  Volume  and  Surface  Integrals  in  Matlie- 


THE   FIRST   LAW  OF   THERMODYNAMICS          39 

matical  Physics,  Cambridge)  or  a  macro-differential  (Planck  : 
Adit  Vorlesungen  ilber  tkeoretischc  Physik,  Leipzig,  1910,  3 
Vorles. ;  Theorie  der  Wdrmcstrahlung,  pp.  129  et  seq.). 

We  are  very  often  concerned  with  magnitudes  such  as  pressure, 
density,  concentration,  temperature,  etc.,  which  have  the  signifi- 
cance of  mean  values,  and  it  must  be  remembered  that  we  cannot 
apply  these  terms  to  systems  which  are  so  constituted  as  to  pro- 
hibit the  existence  of  such  a  mean  value.  This  point  is  by  no 
means  merely  a  logical  or  mathematical  refinement,  but  is  of  the 
very  essence  of  the  physical  interpretation  of  the  second  law  of 
thermodynamics  (cf.  Planck,  loc.  cit.). 

21.     Units  of  Pressure. 

unit  force 

unit  pressure      .         .     =  — r—    — . 
unit  area 

1   dvnG 


Absolute  unit  of  pressure  =          a    . 

Statical  units  of  pressure  = — ^-4, ^-,  etc. 

cm.2 '   cm.2  ' 

Standard  Atmosphere  =  pressure  of  a  column  of  76  cm.  pure 
mercury  at  0°C.  in  latitude  45°  (variation  of  gravitation  constant 
affects  the  result  J  per  cent,  per  degree) : 

p  =  76  X  density  =  76  X  13'595  =  1033-2  -^ 

cm.a 

=  1033-2  X  980-53  =  1013130  d^?2e. 
A  standard  C.G.S.  atmosphere  of  106  — — %  has  also  been  proposed. 

The  dimensions  of  pressure  are  : 
force  force 


area       (length)2  " 
22.     Fluids. 

We  may  for  the  purposes  of  thermodynamics  define  a  fluid  as 
follows : 

(1)  The  line  of  action  of  the  thrust  exerted  by  a  fluid  at  rest 
on  an  area  is  everywhere  perpendicular  to  the  area. 

(2)  The  pressure  at  a  point  in  a  fluid  at  rest  is  equally  great  in 
all  directions. 

This  implies  that  a  small  element  of  area  8a  may  be  turned  so 


40  THERMODYNAMICS 

as  to  be  perpendicular  to  every  direction  drawn  through  the 
selected  point  (x,  y,  z),  without  thereby  altering  the  thrust 
upon  it. 

(3)  A  pressure  applied  at  any  point  on  the  boundary  of  a  fluid  is 
transmitted  uniformly  throughout  the  whole  fluid  (Pascal's  law). 

In  what  follows  we  shall  always  consider  the  pressure  as  having 
a  uniform  value  for  all  directions  through  any  point.  Gases  and 
liquids  at  rest  satisfy  this  condition  ;  under  some  circumstances 
a  solid  may  be  treated  thermodynamically  as  a  "  fluid,"  e.g., 
when  it  is  immersed  in  a  liquid  under  pressure  and  is  free  from 
torsion  or  shearing  stress.  Conditions  (2)  and  (3),  however,  very 
materially  limit  the  range  of  applicability  in  such  cases. 

23.     Elasticity  of  a  Fluid. 

The  modulus  of  elasticity  of  volume,  or  bulk  modulus  of  elasticity, 
of  a  fluid  under  specified  conditions,  is  the  ratio  of  any  small 
increase  of  pressure  to  the  resulting  relative  decrease  of  volume. 
We  shall  refer  to  this  simply  as  the  elasticity. 

Let  p  and  r  be  the  initial  pressure  and  volume,  and  suppose  p 
is  increased  to  (p  +  Bp),  the  total  volume  thereby  increasing  to 
(v  +  8v).  Then  : 

increase  of  pressure  =  Bp, 

increase  of  volume  =  Sr, 

relative  increase  of  volume  =  Bv/v, 

Bp  Bp  dp  .    ,,     ,.    .. 

Elasticity  =  e  =  —  ~-  =  —  v  ~-  =  —  v  -r ,  in  the  limit. 
Bv/v  Be  dv 

The  negative  sign  indicates  that  all  real  fluids  contract  when 
the  pressure  increases.  Since  Bv/r  is  a  mere  number,  the 
elasticity  has  the  same  dimensions  as  pressure. 

The  reciprocal  of  the  elasticity  of  volume  of  a  fluid  is  called  its 
modulus  of  compressibility  (r?)  : 

dv 
--. 

dp 

It  is  easily  shown  that,  if  corresponding  values  of  v  and  p  are 
represented  in  a  rectangular  co-ordinate  system,  the  elasticity  at 
any  point  on  the  curve  is  equal  to  the  length  of  the  _p-axis  inter- 
cepted between  the  tangent  at  that  point  and  the  horizontal 
through  the  point  (Fig.  1). 


tnroi 


THE   FIRST   LAW  OF   THERMODYNAMICS 


11 


The  elasticity  of  a  fluid  is  not  completely  specified  unless  the 
conditions  under  which  it  is  measured  are  known,  and  a  fluid 
has  various  elasticities  corresponding  to  the  different  sets  of 
conditions.  Two  are  especially  important : 

(1)  Isothermal  elasticity,  fe,  measured  under   such  conditions 
that  the  temperature  remains  constant. 

(2)  Adiabatic  elasticity,  eQ,  measured  under   such   conditions 
that  no  heat  is  allowed  to  enter,  or  escape  from,  the  fluid  during 
the  volume  change. 

In  the  measurement  of  the  former,  the  cylinder  enclosing  the 
fluid  may  be  supposed  to  be  formed  of  a  good  conductor  of  heat, 
and  to  be  immersed  in  a  large  tank  of  water  at  the  particular 
temperature ;  if  the  compres- 
sions are  effected  very  slowly, 
the  mass  of  fluid  will  then 
have  the  opportunity  of  acquir- 
ing, at  every  instant,  the  tem- 
perature of  its  surroundings 
by  heat  transfer,  so  that  it  will 
remain  at  a  constant  tempera- 
ture. The  adiabatic  elasticity, 
on  the  other  hand,  applies  to 
compressions  of  a  fluid  en- 
closed in  a  perfectly  non-con- 
ducting cylinder,  or,  since  no 
actual  cylinders  satisfy  this 
condition,  to  pressure  changes  which  are  performed  very  rapidly, 
so  that  there  is  not  sufficient  time  allowed  for  equalisation  of 
temperature  by  heat  transfer.  An  example  of  such  pressure 
changes  is  afforded  by  the  periodic  variations  of  pressure  at  a 
point  in  a  gas  which  is  transmitting  a  succession  of  sound-waves. 

24.     Work  done  by  an  Expanding  Fluid. 

If  a  fluid  contained  in  a  cylinder  expands  so  that  its  pressure 
remains  constant  (e.g.,  saturated  steam  in  contact  with  water),  the 
work  done  is  that  of  raising  the  piston,  of  area  a,  which  supports 
a  weight  W  just  sufficient  to  keep  the  expansive  force  indefinitely 
near  equilibrium.  If  s  =  distance  of  outward  motion  of  piston  : 
work  A  =  W.  s  =  pa.  s  =  p.  as  =  p&u,  where  Ar  is  the  increase 
of  volume. 


FIG.  i. 


42 


THERMODYNAMICS 


It  is  easy  to  show  that  this  relation  is  perfectly  general.  Let 
a  mass  of  fluid  of  any  shape  be  represented  (Fig.  2)  by  the  full 
line  PQRS,  and  let  its  volume  change,  under  a  uniform  external 
pressure  p  acting  everywhere  normal  to  the  bounding  surface,  so 
that  the  final  volume  is  represented  by  the  dotted  line  PQ'R'S'. 
We  now  take  a  small  element  of  area  on  the  original  surface  and 
erect  upon  it  a  cylindrical  surface  C,  cutting  the  surface  PQ'R'S'. 
Let  the  volume  enclosed  between  the  two  elements  of  area  on  the 
surfaces  bounding  the  cylindrical  space  be  Ar ;  this  is  positive  if 
the  surface  PQ'R'S'  lies  outside  the  surface  PQRS  at  the  position 
considered  (as  at  Q'),  negative  if  inside  (as  at  R').  The  work 
done  in  the  infinitesimal  cylinder  is 
therefore  pAr. 

If  the  whole  change  of  volume  is 
divided  into  such  cylinders,  the  total 
work  done  =  SpAi-  =  jjSAr  =  p  X 
(total  change  of  volume)  as  before. 

If  the  pressure  does  not  remain 
uniform  during  the  expansion,  we 
imagine  the  process  divided  into  a 
very  large  number  of  very  small 
changes' of  volume,  in  each  of  which 
the  pressure  may  be  regarded  as 
constant.  Let  pv  be  the  mean  pressure  during  an  expansion 
from  a  volume  (v  —  ^Sr)  to  a  volume  (v  -f-  £Sr) ;  then  the  small 
element  of  work  done  is  : 

BA.  =  _2>c8f. 

For  a  finite  change  of  volume  under  specified  conditions  (e.g., 
constant  temperature) : 


FIG.  2. 


~r2  - 

=      pdv  =  2Wa  —  pivi  — 
J  1-1  ^ 


i  —      vdp. 
pi 


25.     Heat  Function  at  Constant  Pressure. 

The  heat   absorbed   in  the  change  of   state  of  a  system  at 
constant  volume  is  equal  to  the  increase  of  intrinsic  energy  : 

Q^Ua-Ux         .         .        ..          •  (1) 


THE   FIEST  LAW  OF   THERMODYNAMICS          43 

The  heat  absorbed  in  the  change  at  constant  pressure  is  this 
plus  the  external  work  : 

QP  =  U2  -  Ui  +  A  =  U2  -  Ui  +  X'-a  -  t-i)       .      (la) 
The  equation  (la)  may  be  written  in  the  form : 

Q  =  (U2  +  jwra)  -  (Ui  +  pri)  .         .  (16) 

If  we  put  U+j>r=W         ....   (2), 

we  see  that  the  heat  absorbed  when  a  system  passes  from  one 

state  to  another  at  constant  pressure  is  equal  to  the  difference  in 

the  values  of  a  f unction  (U  -\-pi~)  for  the  initial  and  final  states  : 

Qp  =  Wa-Wi     ....     (3) 

The  function  W  was  called  by  Willard  Gibbs  the  Heat  Function 
at  Constant  Pressure. 

Corollary.  f?W  is  a  perfect  differential  when  the  pressure  is 
constant,  and  Qp  is  independent  of  the  path.  The  independence 
of  the  heat  effect  on  the  path  requires  that  the  change  shall  occur 
either  at  constant  volume  or  at  constant  pressure.  If  the  volume 
is  maintained  constant  (dv  =  o)  the  pressure  may  be  changed  in 
any  way ;  if  the  pressure  is  maintained  constant  (dp  =  o)  the 
volume  may  be  altered  in  any  manner  so  that  the  limiting  con- 
ditions are  satisfied;  but  if  both  pressure  and  volume  change 

simultaneously  I  j>dv  is  no  longer  independent  of  the  path. 

26.     Characteristic   Equation. 

It  is  usual  in  physics  and  chemistry  to  speak  of  the  "  state  "  of 
a  given  body,  and  we  may  perhaps  define  the  term  by  saying 
that  two  bodies  are  in  the  same  state  when  they  are  identical 
except  as  regards  accidental  properties  such  as  shape,  position, 
and  size.  The  independence  of  state  on  the  size  implies  that 
when  we  have  defined  the  state  for  unit  mass,  we  have  fixed  it 
for  any  mass.  If  we  abstract  these  unessentials  we  are  left 
with  the  concept  of  a  substance  (cf.  H.  M.,  Introduction).  What 
properties  of  two  portions  of  a  substance  must  agree  in  order  that 
they  shall  be  identical,  i.e.,  in  the  same  state?  In  the  case  of  a 
fluid,  the  following  properties  of  unit  mass  must  be  identical : 

(1)  Chemical  composition. 

(2)  Temperature,  0. 

(3)  Volume,  i.e.,  specific  volume,  r. 


44  THEKMODYNAMICS 

The  pressure  is  then  defined,  and  there  must  be  some  relation 
between  p,  v,  and  6  : 

f(p,v,0)=0       .         .         .         .      (1) 

The  magnitudes  p,  v,  6  are  therefore  determined  by  the  state 
of  the  body,  and  so  may  be  called  functions  of  the  state. 

The  relation  (1)  is  called  the  characteristic  equation  ("  Zustands- 
gleichung  ")  of  the  fluid.  Thus,  if  two  of  the  variables  p,  v,  6  are 
given  certain  values,  the  third  is  fixed,  and  the  state  of  a  homo- 
geneous fluid  is  like  the  position  of  a  point  in  a  plane,  capable 
of  two  and  only  two  independent  variations,  or  has  two  degrees  of 
freedom.  A  particular  point  in  a  plane  may  be  associated  with 
every  separate  state  of  which  the  body  is  capable,  so  that  states 
differing  infinitesimally  are  associated  with  points  infinitely  close 
together,  and  the  assemblage  of  points  may  be  regarded  as  repre- 
senting the  various  states.  Such  a  method  of  association  is  called 
a  continuous  association.  All  the  points  associated  with  states 
of  equal  volume,  pressure,  or  temperature  form  lines,  each  corre- 
sponding to  a  particular  volume,  pressure,  or  temperature,  and 
these  are  called  isochores,  isopiestics,  and  isotherms,  respectively. 

When  two  of  the  magnitudes  v,p,0  have  been  assigned,  the  value 
of  the  third  is  found  by  solving  (1).  Thus,  if  v,  6  are  fixed,  we 
have: 

p  =  <Kr,  6}. 

It  may  happen  that  there  are  two  (or  more)  values  of  the  dependent 
variable  for  one  pair  of  values  of  the  independent  variables.  Thus,  a  liquid 
exhibiting  a  maximum  density  (e.g.,  water  at  4°  C. )  will  have  at  least  two 
values  of  6,  on  opposite  sides  of  this  state,  for  given  values  of  v  and  p.  A 
plane  diagram,  therefore,  does  not  always  adequately  represent  the  states  of 
such  a  fluid. 

If  the  body  changes  its  state,  the  points  associated  with  the 
states  successively  assumed  by  the  body  will  form  a  line  on  the 
diagram,  called  the  path  of  the  body.  During  the  changes,  a 
definite  quantity  of  heat  is  absorbed,  and  a  definite  quantity  of 
work  is  done,  by  the  body.  These  magnitudes,  depending  on  the 
path,  we  may  call  the  heat  and  work  of  the  path.  If  the  changes 
constitute  a  cycle,  the  path  is  a  circuit,  and  if  x,  y  are  the  two 
variables  taken  as  co-ordinates,  the  area  of  the  circuit  is  given 

by  \ydx,  or  \xcly,  according  to  the  convention  adopted  as  to  the 
sign  of  its  area  (cf.  H.  M.,  §§  110—113). 


THE   FIRST  LAW  OF   THERMODYNAMICS 


45 


27.     Indicator  Diagram. 

The  best  known  application  of  the  graphical  method  is  that  in 
which  the  variables  are 

x  =  v  (specific  volume). 
y  —  p  (pressure). 

Any  continuous  change  of  volume  is  represented  (Fig.  3)  by 
a  curve  AB,  and  the  work  done  by  an  infinitesimal  expansion  Si- 
is  represented  by  a  narrow  rectangle  with  base  Si-  and  mean 
altitude  p,.  The  work  done  in  a  finite  expansion  from  volume  i\ 
to  volume  r2  is  therefore  the  area  enclosed  between  the  r  axis,  the 
ordinates  r  =  i\,  and  r  =  ?-2,  and  p 
the  part  of  the  curve  intercepted 


between  them,  i.e., 


(cf.  §  24). 


PI 


\ 


A  quantity  of  work  is  therefore 
represented  by  an  area.  This  area 
is  positive  if  the  tracing  point  runs 
along  the  curve  from  left  to  right 
(expansion) ;  negative  if  from  right 
to  left  (compression). 

The  graphical  representation  of  work 
done  by  an  expanding  fluid  as  an  area  is 
due  initially  to  James  Watt,  who  applied  it  to  the  indication  of  steam  engines  ; 
it  was  generalised  and  introduced  as  a  very  convenient  aid  to  theruiodyna- 
mical  reasoning  by  E.  Clapeyron  in  1834. 

There  is  no  limitation  imposed  on  the  motion  of  the  tracing 
point,  save  that  this  must  be  continuous  and  confined  to  the 
part  of  the  (p,r)  plane  lying  to  the  right  of  the  axis  of  p, 
negative  volumes  having  no  physical  significance. 

The  value  of  the  work  is  therefore  dependent  on  the  shape  of 
the  whole  of  the  curve  between  the  initial  and  final  volumes,  and 
hence,  in  determining  the  work  done  in  tracing  out  an  element 
of  the  curve  we  must  know  the  direction  of  that  element,  or  in 
other  words  the  elasticity  of  the  fluid.  This  of  course  is  merely 
a  special  case  of  the  theorem  that  8  A  is  in  general  an  imperfect 
differential. 

If  we  suppose  that  the  tracing  point,  after  moving  in  any  con- 


46 


THERMODYNAMICS 


sistent  manner,  returns  finally  to  its  original  position,  the  curve 
obviously  becomes  a  circuit,  and  the  fluid  has  undergone  a  cycle 
of  operations,  the  representation  of  which  on  the  indicator 
diagram  may  assume  various  forms  : 

(1)  If  the  tracing  point  moves  out  along  any  curve  (Fig.  4) 
from  the  initial  position  a^,  p^,  to  the  final  position  /3(r2,  p^ 
and  then  returns  along  the  same  curve  the  work  done  is  zero, 
since  the 


work  along  aft  =  area 

work  along  /3a  =  area  Paviv*  =  —  area  o/3ravi. 

This  is  equivalent  to  the  statement  : 


/»^!2  /V^l 

pdv  —  I  pdv  =  0, 


in  which  p  is  the  same  function  of  v  in  both  integrals. 

(2)  If  the  tracing  point  returns  along  a  different  curve,  the 


FIG.  4. 


FIG.  5. 


representation  of  the  cycle  is  a  closed  loop,  or  several  loops,  each 
enclosing  a  finite  area.  If  the  tracing  point  traverses  any  loop 
in  a  clockwise  sense,  the  area  of  that  loop  is  positive,  since  the 
positive  area  lying  beneath  the  upper  curve  of  the  loop  exceeds 
the  negative  area  lying  beneath  the  lower  curve.  If  the  tracing 
point  describes  any  loop  in  a  counter-clockwise  sense,  the  area 
of  that  loop  is  negative,  for  a  similar  reason.  The  physical 
interpretation  of  this  result  is  clear  if  we  remember  that  in  the 


THE    FIKST   LAW  OF   THEKMODYXAMICS          47 

first  case  the  fluid  is  exposed  to  a  greater  external  pressure  at 
any  given  volume  during  the  expansion  than  it  is  during  the 
compression.  In  the  second  case  the  reverse  is  true. 

If  the  circuit  is  made  up  of  two  or  more  loops,  the  total 
external  work  done  is  the  algebraic  sum  of  the  areas  of  the  loops. 

If  we  consider  a  circuit  made  up  of  two  curves  AaB,  B/2A 
(Fig.  5),  the  area  of  the  loop  is 


Now  consider  the  curve  AySB,  representing  an  expansion  from 
A  (I'ltpi)  to  B  (r2,  ^2)  along  the  curve  p  =f(v).  The  work  of  the 
path  is  represented  by  the  area 


•iA 


Ji'2 
r 

-  n 
j  * 


PI 

(a)  If  the  path  is  a  circuit  p\  =  p%,  vt  =  r2, 


.-.  (\)pdv  =  —  (j)  vdp  quite  generally. 

(b)  The  only  case  in  which  this  relation  is  true  for  an  open 
path,  e.g.,  A/3B,  is  that  of  a  fluid  having  the  characteristic 
equation  pi-  =  constant. 


Here  ^i  =  p*-*  .'.  (I  )  pdv  =  —  (I  )  vdp. 


Such  a  fluid  is  an  ideal  gas  undergoing  isothermal  change  of 
volume  (cf.  §  77). 

28.    Units   of  Work. 

The  unit  in  which  the  work  of  expansion  of  a  fluid  is  expressed 
will  depend  on  the  units  adopted  for  p  and  i:      Its  dimensions 

r;»s2~l 
are  those  of  energy,  viz.,  I  —r 

If  p  is  in  dynes  per  cm.2,  and  v  in  cm.3 


P  X  v=  X  [cm.]»=  [dyne]  X  [cm.]  =  [erg]. 


48  THERMODYNAMICS 

If  p  is  in  grams  weight  per  cm.2,  v  in  cm.3 
pv  =  [gr.  wt.]  X  [cm.]. 

A  convenient  unit  for  gaseous  expansion  is  obtained  by 
measuring  p  in  standard  atmospheres  and  v  in  litres ;  the  work 
done  by  expansion  through  a  volume  of  1  litre  under  a  constant 
pressure  of  one  atmosphere  is  called  a  litre-atmosphere  (I.  aim.'). 
Its  value  in  ergs  or  other  units  may  be  calculated  as  follows  : 

p  =  l  atm.  =  1013130  dyne  per  cm.2, 

v  =  l  litre  =  1000  cm.3, 

.-.  p   X   v  =  1.  atm.  =  1013130  X   1000  erg  =  101313  X 

106  erg. 

But  1  cal.      =  4188  X  107  erg 

/.    1  1.  atm.  =  24191  cal.,  or 
1  cal.       =  0-0413  1.  atm. 
Again,  1  atm.  =1033 '2  gr.  wt.  per  cm.2  at  sea-level  and 

latitude  45°, 
.  • .  1  1.  atm.  =  1033-2  X  103  =  1033200  gr.  cm. 

29.     Reversibility   and  Irreversibility  of  Processes. 

A  process  which  can  be  performed  backwards  so  that  all 
changes  occurring  in  any  part  of  the  direct  process  are  exactly 
reversed  in  the  corresponding  part  of  the  reverse  process,  and 
no  other  changes  are  left  in  external  bodies,  is  called  a  reversible 
process. 

No  real  processes  are  reversible ;  the  irreversibility,  however, 
may  be  either  inherent  in  the  process,  or  adventitious.  Processes 
which  cannot,  even  approximately,  be  reversed  by  any  means  we 
possess  may  be  called  Intrinsically  Irreversible  Processes ;  those 
which  can  be  made  to  approach  more  and  more  closely  to  revers- 
ible processes,  by  a  suitable  modification  of  the  conditions 
under  which  they  occur,  may  be  called  Conditionally  Irreversible 
Processes. 

The  production  of  heat  by  friction,  the  passage  of  heat  from 
one  body  to  another  at  a  lower  temperature  by  conduction  or 
radiation,  and  the  diffusion  of  material  substances,  are  intrin- 
sically irreversible  processes.  A  process  (Carnot's  cyclic  process) 
will  be  described  in  the  next  chapter  by  means  of  which  heat 
generated  by  friction  may  be  withdrawn  and  reconverted  into  work. 


THE    FIRST  LAW  OF   THERMODYNAMICS          49 

Only  a  part  of  the  heat  may,  however,  be  so  transformed,  the  rest 
passing  into  a  body  which  did  not  previously  contain  it.  The 
work  obtained  is  thus  necessarily  less  than  that  spent  in 
the  friction.  Again,  heat  passing  into  a  body  by  conduction  or 
radiation  cannot  be  returned  without  leaving  changes  in  other 
bodies.  The  entire  quantity  of  heat  may  it  is  true  be  restored 
to  the  hotter  body,  but  only  if  work  is  at  the  same  time  spent, 
and  the  heat  produced  from  it  forms  part  of  that  restored  to  the 
hot  body.  Such  processes  are  therefore  intrinsically  irreversible. 

As  an  example  of  conditional  irreversibility  may  be  taken  the 
expansion  of  a  gas.  Work  is  done  by  the  change  of  bulk  in 
opposition  to  the  external  forces,  and  heat  is  absorbed  from  the 
environment.  The  conditions  which  must  hold  in  order  that  the 
process  actually  occurs  are  : 

(i.)  The  pressure  of  the  gas  must  always  be  slightly  greater 
than  the  external  pressure. 

(ii.)  The  temperature  of  the  environment  must  always  be 
slightly  greater  than  the  temperature  of  the  system. 

To  reverse  the  sense  of  the  change,  so  that  a  contraction 
replaces  an  expansion,  and  an  emission  of  heat  replaces  an 
absorption  of  heat,  the  pressure  and  temperature  differences  must 
be  reversed ;  so  that  a  process  cannot  proceed  equally  well  in  either 
direction  under  exactly  the  same  conditions.  If  the  differences  are 
made  smaller  and  smaller,  the  only  effect  is  to  prolong  the  dura- 
tion of  the  change,  and  in  the  limit  when  the  pressures  and 
temperatures  of  the  system  and  environment  are  equal  the  pro- 
cess goes  on  reversibly,  so  that  an  infinitesimal  change  of  pressure 
or  temperature  external  to  the  system  reverses  the  direction  of 
change  at  any  part  of  the  process.  Any  state  of  the  system 
is  now  an  equilibrium  state,  and  the  duration  of  the  pro- 
cess is  infinite,  because  an  equilibrium  state  does  not  pass 
into  another  state.  A  reversible  process  consisting  as  it  does  of 
a  succession  of  states  of  equilibrium  is  not  a  real  process  at  all ; 
it  is  an  ideal  limiting  case,  to  which  real  processes  may  approach 
more  and  more  closely,  but  never  actually  attain.  We  may 
however  consider  the  reversible  process,  not  as  a  continuous 
sequence  of  states  of  absolute  equilibrium,  but  rather  as  a  con- 
tinuous sequence  of  states  indefinitely  near  to  equilibrium  states, 
so  that  an  infinitesimal  change  of  an  external  condition  will,  at 
any  part  of  the  process,  reverse  its  direction.  Thus  if  the 


50 


THERMODYNAMICS 


P+&P 


p+dp 


\" 


T+6T 


external  pressure  is  always  only  infi  nitesimally  different  from  the 
pressure  of  the  system,  a  change  of  ±8p  will  convert  an  expan- 
sion into  a  contraction  or  vice  versa,  and  if  the  temperature 
of  the  environment  is  only  very  slightly  different  from  that  of 
the  system,  a  change  of  +ST  will  alter  an  absorption  into  a  rejec- 
tion of  heat,  or  vice  versa  (Fig.  6).  The  larger  these  differences, 
the  more  rapidly  the  process  occurs,  and  the  greater  will  be  the 
amount  of  irreversibility  inherent  to  it.  It  may  be  observed  that 
conditionally  irreversible  processes  are  always  accompanied  by 

some  kind  of  friction  or  heat 
conduction,  and  therefore  by  a 
certain  amount  of  intrinsic  irre- 
versibility. Thus  if  the  pressure 
difference  is  large,  the  expan- 
sion is  accompanied  by  rapid 
motions,  the  kinetic  energy  of 
which  is  frittered  down  into  heat 
by  viscous  friction  and  this  heat 
is  then  dissipated  by  conduction 
or  radiation.  In  the  limit  the 
intrinsically  irreversible  changes  vanish  along  with  the  condi- 
tionally irreversible  changes  which  are  their  cause. 

That  all  actual  processes  are  irreversible  does  not  invalidate 
the  results  of  thermodynamic  reasoning  with  reversible  processes, 
because  the  results  refer  to  equilibrium  states.  This  procedure  is 
exactly  analogous  to  the  method  of  applying  the  principle  of 
Virtual  Work  in  analytical  statics,  where  the  conditions  of  equili- 
brium are  derived  from  a  relation  between  the  elements  of  work 
done  during  virtual,  i.e.,  imaginary,  displacements  of  the  parts  of 
the  system,  whereas  such  displacements  are  excluded  by  the 
condition  of  equilibrium  of  the  system. 

Irreversibility  is  considered  more  in  detail  in  §  48. 


T+6T 


FIG.  6. 


CHAPTER  in 

THE  SECOND  LAW  OF  THERMODYNAMICS;  ENTROPY 

30.    The  Transformation  of  Heat  into  Work. 

A  given  quantity  of  any  other  form  of  energy  can,  by  suitable 
available  means,  be  completely  transformed  into  heat. 

The  correctness  of  this  statement  is  to  be  inferred  from  the 
exact  agreement  between  the  values  of  the  mechanical  equivalent 
of  heat  obtained  by  different  methods.  Thus,  in  Joule's 
second  series  of  experiments,  mechanical  work  is  directly  con- 
verted into  heat ;  in  the  first  and  third  series,  it  is  indirectly 
transformed  through  the  medium  of  electro-magnetic  energy :  in 
the  fourth  series,  the  energy  of  an  electric  current  is  converted 
into  heat :  the  identity  of  the  values  of  J  so  obtained  implies  a 
complete  conversion  of  the  initial  forms  of  energy  into  heat  energy. 

That  the  reverse  conversion  of  heat  into  work  is  at  least  partially 
possible  is  established  by  the  existence  of  heat  engines.  We  have 
therefore  to  determine  if  a  given  quantity  of  heat  can  be  com- 
pletely converted  into  work  by  such  engines,  i.e.,  if  the  process  can 
be  aschistic,  and  if  not,  what  is  the  relation  between  the  total 
quantity  of  heat  and  the  part  transformed  into  work.  Let  us  for 
a  moment  suppose  that  such  an  aschistic  engine  exists.  It  would 
then  suffice  to  sink  this  apparatus  in  a  large  heat  reservoir, 
such  as  the  ocean,  to  obtain  immeasurably  large  quantities  of 
work.  Such  an  engine  would  rotate  the  screw  of  a  ship  by 
drawing  on  the  heat-content  of  the  sea,  and  since  the  work  done 
on  the  screw  is  ultimately  reconverted  into  heat  by  friction  the 
engine  could  work  for  ever,  using  up  heat  as  fast  as  it  produced 
it.  Another  type  of  such  an  engine  would  be  a  machine  which  is 
driven  at  the  expense  of  the  heat  generated  by  friction  in  its  own 
moving  parts.  There  is  nothing  in  the  first  law  violated  by  such 
arrangements,  because  there  is  no  creation  of  energy,  and  the 
appliances  are  not  perpetua  mobilii  of  the  first  kind.  Nevertheless, 
the  feasibility  of  making  such  wonderful  engines  would  be  at  once 

E  2 


52  THERMODYNAMICS 

denied  by  practical  engineers,  and  we  may  regard  it  as  a  fact  of 
experience  that  these  appliances  are  improbable  in  the  highest 
degree — that  is,  impossible.  If  we  call  such  an  arrangement, 
capable  of  converting  unlimited  quantities  of  heat  at  constant 
temperature  aschistically  into  work,  a  perpetuum  mobile  of  the 
second  kind,  we  may  assert  that  their  existence  is  in  contradiction 
to  some  generalisation  of  the  results  of  experience — that  is,  to 
some  natural  law.  This  is  the  Second  Law  of  Thermodynamics. 

31.     The  Second  Law  of  Thermodynamics. 

It  is  impossible  to  construct  a  machine  which  shall  work  in  a  cycle 
and  produce  no  effects  other  than  the  cooling  of  a  heat  reservoir  and 
the  raiting  of  a  weight. 

By  "  cooling  "  we  understand  "  abstraction  of  heat,"  and  the 
statement  is  therefore  equivalent  to  the  denial  of  the  possibility 
of  constructing  an  appliance  which  continually  absorbs  heat,  and 
gives  out  work  without  producing  any  other  changes  in  surround- 
ing bodies.  The  question,  "  To  what  extent  is  conversion  of  heat 
into  work  possible  ?  "  has  therefore  the  answer,  "  To  any  extent 
which  does  not  violate  the  Second  Law  of  Thermodynamics."  In 
particular  it  should  be  possible  to  obtain  work  from  a  given 
quantity  of  heat  in  an  arrangement  such  that  some  heat  passes 
into  surrounding  bodies ;  the  rise  of  temperature  of  these  will 
constitute  a  "  change,"  and  this  is  an  "  effect  "  produced  in  addi- 
tion to  the  cooling  of  the  initial  heat  reservoir  and  the  per- 
formance of  work  (which  may  always  be  applied  to  the  raising 
of  weights). 

We  have  purposely  introduced  the  word  "  machine,"  implying 
thereby  a  contrivance  of  such  dimensions  as  to  be  capable  of 
construction,  at  least  imperfectly,  from  available  systems.  We  can 
in  fact  imagine  various  arrangements  which  would  bring  about 
conversion  of  heat  into  work  in  a  manner  directly  contradictory  to 
the  second  law.  All  such  imaginary  appliances,  however,  possess 
one  important  characteristic  :  they  are  (by  reason  of  their  minute 
dimensions)  absolutely  beyond  the  control  of  conscious  beings  of 
our  own  size,  and  possessing  our  faculties,  whilst  such  control  is 
absolutely  necessary  to  the  working  of  the  appliance.  The  second 
law  is  therefore  like  the  first  law  based  ultimately  on  experience 
and  is  not  axiomatic. 


THE    SECOND  LAW  OF   THERMODYNAMICS        53 

The  question  as  to  whether  it  can  be  enunciated  in  such  a  way 
that  the  specific  subjective  content  is  eliminated  will  be  taken  up 
later.  The  form  adopted  above  is  due  to  Planck. 

32.     Heat  Engines. 

A  heat  engine  is  any  arrangement  which  converts  heat  into 
work. 

Steam,  gas,  petrol,  and  hot-air  engines  are  heat  engines  ;  a 
thermopile  coupled  with  an  electromotor  also  constitutes  a  heat 
engine.  An  electromotor  is  not  a  heat  engine,  since  its  effect  is 
produced  at  the  cost  of  electrical  energy,  which,  may,  it  is  true, 
ultimately  be  derived  from  a  heat  engine  coupled  with  a  dynamo, 
but  may  equally  well  arise  from  chemical  action  in  voltaic  cells 
absorbing  practically  no  heat  from  their  environment  (e.g.,  the 
Daniell  cell). 

A  heat  engine  consists  of  three  essential  parts  : 

(a)  A  source,  or  hot  reservoir,  from  which  heat  is  absorbed  by 

the  engine  ; 

(b)  A  working  substance    (steam,    air,    etc.)   which    undergoes 

changes  (e.g.,  volume  changes)  whereby  a  part  of   the 
heat  absorbed  from  the  source  is  converted  into  work  ; 

(c)  A  refrigerator,  or  cold  reservoir,  which  receives  from  the 

working  substance  that  part  of  the  heat  absorbed  from 
the  source  which  is  not  converted  into  work. 

Efficiency  of  a  Heat  Engine.  —  The  efficiency  (X)  of  a  heat 
engine  is  measured  by  the  fraction  of  the  quantity  of  heat 
received  from  the  source  which  is  converted  into  work,  both  heat 
and  work  being  measured  in  the  same  units. 

Let  the  engine  take  Qi  units  of  heat  from  the  source,  produce 
A  units  of  work,  and  give  up  Q2  units  of  heat  to  the  refrigerator  ; 
then  the  efficiency  is  N  =  A/Q!  .....  (a) 

If  the  conversion  of  heat  into  work  is  aschistic  : 

A  =  Q!  -  Q2 


..  .      .... 

Equation  (a)  (being  a  definition)  is  generally  applicable  ;  equa- 
tion (b)  only  to  aschistic,  in  particular  to  cyclic,  processes. 

The  best  modern  heat  engines  (steam  cylinder,  turbine,   or 


54  THERMODYNAMICS 

internal  combustion)  transform  only  about  one-third  of  the  heat 
energy  into  work. 

33.  Reversible  Engines. 

Definition. — If  an  engine  is  such  that,  when  it  is  worked 
backwards,  the  thermal  and  mechanical  effects  in  every  part  of 
its  motions  are  all  reversed,  it  is  called  a  reversible  engine. 

Corollary  (1).  The  processes  involved  in  the  operations  of  a 
reversible  engine  are  all  reversible  processes. 

Corollary  (2).  The  conditions  to  be  satisfied  for  reversibility 
are  : 

(1)  The   temperature  of   the   working   substance  must   never 
differ  more  than  infinitesimally  from  that  of  any  body  with  which 
it  comes  in  contact,  otherwise  irreversible   transfer  of   heat  by 
conduction  or  radiation  occurs. 

(2)  The  pressure  at  every  instant  during  an  expansion  or  con- 
traction of  the  working  substance  must  be  only  infinitesimally 
greater  or  less  respectively,  than  the  external  pressure,  other- 
wise turbulent  motions  occur,  the  kinetic  energy  of  which  is  ulti- 
mately converted  into  heat  by  friction,  and  this  heat  production 
is  intrinsically  irreversible. 

(3)  Friction  at  all  moving  parts  must  be  absent,  otherwise  irre- 
versible production  of  heat  is  involved  both  in  the  forward  and 
backward  working  of  the  engine. 

34.  Camel's  Cycle. 

The  problem  of  the  conversion  of  heat  into  work,  although  it 
had  received  a  very  satisfactory  practical  solution  in  the  invention 
and  improvement  of  the  steam  engine,  did  not  form  the  subject 
of  any  theoretical  investigation  until  Sadi  Carnot,  in  1824, 
published  his  "  Reflexions  sur  la  puissance  motrice  du  jen,  et  snr 
les  machines  propres  d  developer  cette  puissance"  The  funda- 
mental problem  was  stated  by  Carnot  with  great  clearness.  "  To 
examine  the  principle  of  the  production  of  motion  [work]  by 
heat  in  all  its  generality,  it  must  be  conceived  independently  of 
any  mechanism,  or  of  any  particular  agent ;  it  is  necessary  to 
establish  proofs  applicable  not  only  to  steam  engines,  but  also 
to  other  heat  engines,  irrespective  of  the  working  substance  and 
the  manner  in  which  it  acts." 


THE    SECOND   LAW  OF   THERMODYNAMICS         55 

With  this  aim,  Carnot  proceeded  to  introduce  the  novel  concep- 
tion of  a  reversible  cycle  of  operations,  and  arrived  at  the 
exceedingly  important  result  that  "  the  motive  power  [i.e., 
capacity  for  doing  work]  of  heat  is  independent  of  the  agents 
employed  to  develop  it ;  its  quantity  is  determined  solely  by  the 
temperatures  of  the  bodies  between  which,  in  the  final  result  [i.e., 
after  the  completion  of  the  cycle],  the  transfer  of  heat  occurs." 

Let  there  be  given  a  source,  and  a  refrigerator,  at  tempera- 
tures TI,  T2  respectively,  where  TX  >  T2.  In  order  that  finite 
quantities  of  heat  may  be  added  to  or  taken  from  these  without 
change  of  their  temperatures,  we  may  suppose  them  to  consist  of 


FIG.  7. 

large  reservoirs  of  water,  or  still  better  of  reservoirs  of  steam  and 
ice,  which  would  preserve  constant  temperatures  during  with- 
drawal or  addition  of  heat. 

Nothing  more  is  assumed  about  the  temperatures,  and  one 
result  of  Carnot's  investigation  is  a  rigorous  definition  of  tempera- 
ture. Further,  let  there  be  a  cylinder  and  piston,  of  an  absolute 
non-conductor  of  heat,  closed  at  the  bottom  by  a  perfect  con- 
ductor of  heat,  and  containing  the  working  substance — any 
substance,  or  mixture  of  substances,  the  pressure  of  which  is 
uniform  in  all  directions  at  all  points  and  is  a  continuous  function 
of  temperature.  Finally,  we  have  a  stand  formed  of  a  perfect 
non-conductor  of  heat  (Fig.  7). 

There  is  now  performed  a  reversible  cycle  called  Carnot's  cycle, 
and  consisting  of  four  operations  : 


56  THERMODYNAMICS 

(1)  The  working  substance  being  initially  at  the  temperature 
T2  of  i he  refrigerator,  we  place  the  cylinder  on  the  non-conducting 
stand,  and  compress  the  working  substance  reversibly  until  the 
temperature  rises  to  1\.     By  the  conditions  imposed,  this  is  an 
adiabatic  compression,  and  will  be  represented  by  a  continuous 
curve  on  the  indicator  diagram,  say  AB  (Fig.  8). 

(2)  We  now  place  the  cylinder  on  the  source,  and  allow  the 
working  substance  to  expand  reversibly  and  isothermally  at  TI 
until  any  arbitrary  quantity  of  heat  Qi  has  been  absorbed. 

In   this  process   the  temperature  of  the  working   substance 

must,  it  is  true,  be  infini- 
tesimally  less  than  that  of 
the  source  in  order  that 
heat  may  pass  into  it,  but 
in  the  limit  this  difference 
becomes  vanish ingly  small, 
-/Q, absorbed  Prom  and  the  temperatures 
approach  equality.  This 
step  is  represented  by  a 
continuous  curve,  the  iso- 
therm BC. 

(3)  The  cylinder  is  again 
Q,  rejected  to  refrig?         kced  Qn  the  non.conduct. 


FlG  8  ing  stand,  and  the  working 

substance  reversibly  and 

adiabatically  expanded  till  its  temperature  falls  to  T2.  The 
course  of  expansion  is  represented  by  the  curve  CD. 

(4)  Finally,  the  cylinder  is  placed  on  the  refrigerator  and  the 
working  substance  compressed  reversibly  and  isothermally  until 
it  returns  to  its  initial  state  A,  rejecting  heat  Q2  to  the  refrigerator. 
This  operation  is  represented  by  the  curve  DA. 

The  cycle  is  now  completed,  and  the  working  substance  is  in 
exactly  the  same  state  as  at  the  beginning.  It  will  be  observed 
that,  with  a  given  initial  state,  and  given  temperatures,  all  the 
curves  are  completely  determined  by  the  length  of  BC,  which 
alone  is  arbitrary.  The  working  substance  is  never  in  contact 
with  bodies  differing  from  it  more  than  infinitesimally  in 
temperature,  and  is  never  exposed  to  pressures  exceeding  or 
falling  short  of  its  own  by  more  than  infinitesimal  amounts. 
The  cycle  is  therefore  reversible,  and  may  be  carried  out  in  the 


THE   SECOND   LAW  OF   THEEMODYNAMICS-       57 


reverse  direction,  the  thermal  and  mechanical  effects  at  every 
part  being  exactly  reversed,  so  that  if  in  any  infinitesimal 
element  of  the  direct  cycle: 

the  difference  of  pressures  is  +  Sp, 
the  difference  of  temperatures  is  +  ST, 
the  heat  absorbed  is  -f-  SQ, 
the  work  done  is  +  $A, 

then  in  the  reverse  .cycle  that  infinitesimal  element  of  path  is 
retraced,  and  all  the  above  magnitudes  have  negative  signs,  so 
that  compressions  correspond  to  expansions,  heat  evolved  to  heat 
absorbed,  work  spent  to  work  done,  and,  to  produce  the  opposite 
direction  of  heat  transfer,  the  temperature  differences  must  also 
be  reversed  _in .  sign.  The 
actual  course  of  the  reversed  P 
cycle  will  therefore  l^e  com- 
posed of  the  four  steps : 

(la)  Starting  at  the  tem- 
perature T2,  expand  rever- 
sibly  and  isothermally  along 
AB  (Fig.  9),  till  an  amount 
of  heat  Q2  is  absorbed  from 
the  refrigerator. 

(2a)  Compress  reversibly 
and  adiabatically  along  BC 
until  the  temperature  rises 
to  Ti. 

(3a)  Compress  reversibly 
and   isothermally  along   CD  until  heat  Qi  is  given  up  to  the 
source. 

(4a)  Expand  reversibly  and  adiabatically  along  DA,  finishing 
at  the  initial  state  A. 

In  the  direct  circle  the  loop  ABCD  is  described  in  a  clockwise 
sense ;  in  the  reversed  circle  it  is  described  in  a  counter-clockwise 
sense. 

We  will  now  consider  the  changes  produced  in  the  direct  and 
reversed  cycles. 

(a)  In  both  cases  the  working  substance  is  unchanged. 

(b) .  In  the  direct  cycle  a  quantity  of  heat  Qi  is  absorbed  from 
the  source,  and  a  quantity  of  heat  Q2  is  given  up  to  the 
refrigerator.  In  the  reversed  cycle  a  quantity  of  heat  Q2  is 


B 

bsorbed  Prom  rePrig!" 


FIG.  9. 


58  THERMODYNAMICS 

absorbed  from  the  refrigerator,  and  a  quantity  of  heat  Qi  is 
rejected  to  the  source. 

(c)  In  the  direct  cycle  an  amount  of  work  A,  represented  by 
the  area  ABCD,  is  done  by  the  system  ;  in  the  reversed  cycle  an 
amount  of  work  —  A  is  done  by  the  system,  i.e.,  the  work  +  A  is 
spent  on  the  system, 

35.    Carnot's  Theorem. 

We  shall  now  apply  the  two  laws  of  thermodynamics  to  the 
energy  changes  occurring  in  the  Carnot's  cycle. 

(1)  The  cyclic  process  being  aschistic,  we  have,  as  a  consequence 
of  the  first  law  : 

Qi-Qa^A         .         .         .         .     (1) 

(2)  For  the  application  of  the  second  law  we  establish  the 
following  propositions  : 

(a)  Of  all  possible  heat  engines  working  with  fixed  temperatures 
of  source  and  refrigerator,  a  reversible  engine  is  the  most  efficient. 

Let  [a],  [/3]  be  two  engines  having  the  same  source  and 
refrigerator ;  let  [a]  be  a  reversible  Carnot's  engine,  [/8]  some 
other  engine  which  (if  possible)  is  more  efficient  than  [a].  By 
suitable  adjustment  of  the  working  parts  (e.g.,  length  of  piston 
stroke)  each  engine  may  be  arranged  so  as  to  absorb  heat  Qi  from 
the  source  per  complete  cycle.  Now  let  [ft]  work  [a]  backwards, 
so  that  [a]  converts  the  work  it  receives  into  heat.  This 
operation  is  possible,  because  [a]  is  a  reversible  engine.  In  a 
complete  cycle  : 

[takes  up  heat  Qi  from  the  source, 
[/3]  j  does  an  amount  of  work,  say  A', 

(gives  up  heat  Qa'  to  the  refrigerator, 
/returns  heat  Qi  to  the  source, 
[a]  j  absorbs  work  A, 

I  takes  up  heat  Q2  from  the  refrigerator. 
The  efficiency  of  [/?]  is,  by  definition, 

N,  =  A'/Qi, 
and  that  of  [a]  is  similarly 

Na  =  A/QL 
By  hypothesis,  Np  >  Na  , 

.-.  A'  >  A         .         .         .         . 
But  A'  +  Qa'  =  A  +  Q2  =  Qt,  by  the  first  law, 

.'.   Qa'  <  Qa        -          .          -          - 


THE   SECOND  LAW  OF   THERMODYNAMICS       59 

Equations  (a)  and  (b)  show  that  the  compound  engine  [a  -f  0] 
is  capable  of  producing  an  amount  of  work  (A'  —  A),  which  could 
be  used  to  raise  a  weight,  and  that  it  leaves  no  other  change 
in  the  surroundings  except  that  the  refrigerator  is  cooled  by 
withdrawal  of  an  amount  of  heat  (Qa  —  Qa')- 

This  result  is  in  contradiction  to  the  second  law,  hence  we 
conclude  that  the  hypothesis  entertained  is  inadmissible,  so  that 
[/3]  is  an  impossible  engine,  which  establishes  the  proposition. 

(b)  All  reversible  engines  -working  in  cycles  with  the  same 
temperatures  of  source  and  refrigerator  are  equally  efficient. 

For  if  [a]  is  a  Carnot's  reversible  engine  with  any  specified 
working  substance,  and  [/3]  another  Carnot's  engine  with  a 
different  working  substance,  or  any  other  reversible  heat  engine 
whatever,  the  preceding  reasoning  shows  that  [/3]  cannot  be  more 
efficient  than  [a].  But  since  [/3]  is  itself  a  reversible  engine,  the 
functions  of  the  two  engines  may  be  interchanged,  and  the  same 
reasoning  shows  that  [a]  cannot  be  more  efficient  than  ~3~ . 
Hence,  since if/8]  cannot  be  more  efficient,  or  less  efficient,  than 
[a],  it  must  Be  equally  as  efficient  as  [a],  so  that  Na  =  X^. 

From  this  we  deduce  the  following  important  corollari/ :  If  a 
quantity  of  heat  Q  is  absorbed  in  a  reversible  cycle,  with  given 
temperatures  of  source  and  refrigerator,  the  quantity  of  work  A 
obtained  from  it  is  independent  of  the  arrangement  used  in 
performing  the  cycle. 

This  we  shall  call  Carnot's  Theorem. 

Beginners  are  usually  surprised  when  they  are  informed  that  the  work 
done  by  a  reversible  engine  performing  a  cycle  between  fixed  temperatures 
is  always  the  same,  for  a  given  quantity  of  heat  absorbed  from  the  source, 
no  matter  what  is  the  working  substance  in  the  engine.  This  may  in  a 
fluid  engiue  be  a  gas  (such  as  air),  a  vapour  (such  as  steam),  a  liquid  (such 
as  water),  a  solid  (such  as  ice  or  copper),  or  a  mixture  of  any  of  these ;  the 
only  condition  imposed  being  that  the  volume  and  pressure  shall  change 
continuously  with  alteration  of  temperature.  Xow  it  would  appear  that  an 
engine  working  with  a  very  volatile  liquid,  say  ether  and  its  vapour,  should 
yield,  fur  a  given  expansion,  much  more  work  than  a  similar  engine  working 
with  water  and  its  vapour,  by  reason  of  the  greater  vapour  pressure  of  the 
former.  "Whilst  this  is  quite  true  under  the  condition  italicised,  yet  it  must 
be  borne  in  mind  that  Carnot's  theorem  applies  only  to  an  engine  which 
works  in  agreement  with  two  very  important  conditions,  viz.,  that  it  works 
in  a  cycle,  and  works  reversibly.  Thus,  although  more  work  is  obtained  in 
the  single  forward  stroke  of  the  ether  engine  than  in  the  similar  stroke  of 
the  steam  engine,  yet,  for  the  same  reason,  proportionally  more  work  must 


60  THERMODYNAMICS 

be  given  back  again  in  the  reverse  stroke  which  completes  the  cycle,  and 
the  greater  loss  exactly  compensates  the  greater  gain. 

36.  Isothermal  Cycles. 

Theorem. — The  work  done  in  any  isothermal  reversible  cyclic 
process  is  zero  (J.  Moutier,  1875). 

For  if  a  cyclic  process  could  be  performed  in  a  heat  reservoir 
of  uniform  temperature  so  as  to  give  out  work,  it  would  consti- 
tute a  perpetuum  mobile  of  the  second  kind,  the  existence  of  which 
is  denied  by  the  second  law.  And  if  the  cyclic  process  absorbed 
work  when  performed  at  a  uniform  temperature,  it  would,  by 
reason  of  its  reversibility,  give  out  an  equal  amount  of  work 
when  reversed  ;  this  would,  however,  be  the  case  first  considered. 
Hence  the  production  of  work  in  either  cycle  is  impossible,  which 
establishes  the  theorem. 

Corollary  1. — The  area  enclosed  by  the  circuit  representing  an 
isothermal  reversible  cycle  on  the  indicator  diagram  is  zero ;  if, 
therefore,  the  curve  is  not  a  segment  of  a  line  transversed  from 
A  to  B  and  then  from  B  to  A  (Fig.  4),  it  must  form  two  loops 
of  equal  areas  but  traversed  in  opposite  senses,  or  else  such 
a  system  of  positive  and  negative  loops  that  the  total  area  is  zero. 

Corollary  2. — The  algebraic  sum  of  the  quantities  of  heat  with- 
drawn from  or  given  to  the  constant  temperature  reservoir  in  an 
isothermal  reversible  cycle  is  zero. 

37.  Absolute  Temperature. 

It  is  an  immediate  consequence  of  Carnot's  theorem  that  the 
ratio  of  the  quantities  of  heat  absorbed  and  rejected  by  a  per- 
fectly reversible  engine  working  in  a  complete  cycle,  depends  only 
on  the  temperatures  of  the  bodies  which  serve  as  source  and 
refrigerator.  .  . 

For  if  Qi,  Q2  are  the  quantities  of  heat  absorbed  and  rejected, 
respectively,  in  localities  at  temperatures  TI,  T2,  then  the 

Efficiency  (N)  =  ^  ~  ^2  =  1  —  9?, 
Vi  ^i 

and  since  N  depends  solely  on  the  temperatures/ by  Carnot's 
theorem, 

\    "~  Q  )  is  a  function  of  TI>  Ta  alone, 
n2  is  a  function  of  TI,  T2  alone, 


THE   SECOND  LAW   OF   THEEMODYNAMICS        61 


or  =  ^(Ta,T!)     .....     (1) 

We  shall  now  suppose  that  nothing  more  is  known  about  the 
temperatures  TI,  T2  except  that  : 

(i.)  A  definite  temperature  may  be  assigned  to  any  material 
system  which  is  in  thermal  equilibrium  ; 

(u.)  The  temperature  of  the  source  is  greater  than  the  tempera- 
ture of  the  refrigerator,  i.e.,  TI  >  T2. 

These  statements  are  implied  in  the  definition  of  temperature 
given  in  §  4. 

Equation  (1)  now  gives,  as  Lord  Kelvin  pointed  out  in  1848, 
a  quantitative  definition  of  temperature,  and  this  definition,  being 
framed  in  terms  of  the  efficiency  of  a  reversible  engine,  is  indepen- 
dent of  the  methods  adopted  for  its  measurement,  and  is  therefore 
"  absolute"  On  the  other  hand,  a  definition  of  the  temperature 
of  a  body  in  terms  of  the  volume  of  a  given  mass  of  air,  mercury, 
etc.,  which  is  in  thermal  equilibrium  with  the  body,  will  be  depen- 
dent on  the  properties  of  some  other  system  than  the  one  of 
which  we  require  to  know  the  temperature  :  in  fact  the  tempera- 
ture so  measured  will  depend  on  the  particular  thermometric 
substance  selected.  If  two  exactly  similar  thermometer  tubes  are 
taken,  one  being  filled  with  mercury  and  the  other  with  alcohol, 
and  if  the  levels  at  which  the  liquids  stand  when  the  tubes  are 
placed  in  melting  ice  and  in  steam  are  marked  0°  and  100  z 
respectively,  each  of  the  100  equal  intervals  between  these  marks  is 
defined  as  a  Centigrade  degree.  But  if  both  thermometers  are  now 
placed  in  thermal  contact  with  some  body  at  a  temperature  inter- 
mediate between  0°  and  100°,  the  readings  of  the  two  thermo- 
meters will  not  agree.  In  order  that  the  temperature  of  a  body 
may  be  defined  without  ambiguity  it  is  necessary,  therefore,  to 
select  a  particular  thermometric  substance  as  standard,  and  to 
use  this  exclusively  in  all  measurements  ;  hydrogen  gas  is  the 
substance  at  present  used  (cf.  §  3).  The  temperature  so  measured 
still  suffers  from  the  defect  that  it  is  not  solely  dependent  on  the 
state  of  the  body  submitted  to  examination,  whereas  the  tempera- 
ture of  a  body  is  undoubtedly  a  function  of  the  state  of  that  body 
alone.  In  the  same  way  the  mass  of  a  body  is  a  property  belong- 
ing solely  to  that  body  :  if,  however,  it  is  estimated  by  a  spring- 
balance,  the  measure  of  the  mass  will  appear  to  depend  on  the 
position  of  the  balance  relative  to  the  earth's  centre  of  gravity. 


62  THERMODYNAMICS 

The  ordinary  balance,  on  the  ..other  hand,  with  suitable  adjust- 
ment of  its  parts,  gives  a  measure  of  the  mass  of  a  body,  in 
terms  of  an  arbitrarily  selected  standard,  which  depends  only  on 
the  particular  body  ;  the  same  measure  would  be  found  if  the 
balance  were  transported  to  any  part  of  the  earth's  surface,  or 
even  to  another  planet,  such  as  Mars  or  Jupiter.  Such  a  measure 
may  be  called  "absolute." 

The  exact  form  of  the  function  $  (T2,  TI)  being  to  some  extent 
arbitrary,  we  might  give  several  definitions  of  "  absolute  tempera- 
ture," all  drawn  up,  however,  in  terms  of  the  efficiency  of  the 
reversible  engine.  Lord  Kelvin,  in  1854,  adopted  the  following 
form  : 

Definition  of  Absolute  Temperature.  —  "  The  temperatures  of  two 
bodies  are  proportional  to  the  quantities  of  heat  respectively 
taken  in  and  given  out  in  localities  at  one  temperature  and  at  the 
other,  respectively,  by  a  material  system  subjected  to  a  complete 
cycle  of  perfectly  reversible  thermodynamic  operations,  and  not 
allowed  to  part  with  or  take  in  heat  at  any  other  temperature  : 
or,  the  absolute  values  of  two  temperatures  are  to  one  another  in 
the  proportion  of  the  heat  taken  in  to  the  heat  rejected  in  a 
perfect  thermodynamic  engine  working  with  a  source  and  refri- 
gerator at  the  higher  and  lower  of  the  temperatures  respectively." 

Hence:  4  (T2,  Tx)  =  ^      .        .        .         .         .     (2), 

where  T  denotes  the  measure  of  the  absolute  temperature  defined 
as  above. 
Equation  (2)  fixes  the  ratio  of  two  temperatures  ;  for  T2/Ti  = 


We  shall  now  define  what  is  to  be  understood  by  equal  intervals 
of  temperature.  Let  us  imagine  that  we  have  a  system  of  rever- 
sible engines  [1,2],  [2,8],  [3,4],  .  .  .  ,  working  between  constant 
temperature  reservoirs  (1),  (2),  (3),  (4),  .  .  .  ,  so  that  the  refrige- 
rator of  any  engine  (except  the  last)  forms  the  source  of  the  next 
engine.  Let  each  perform  a  cycle  so  that 

[1,  2]  takes  heat  Ql  from  (1)  and  gives  out  heat  Q2  to  (2), 
[2,3]     „         „     Q2     „     (2)         „         „         „     Q3,,  (3), 
[3,4]     „         „     Q3     „     (3)         „         „         „     Q4f,  (4), 
and  so  on. 

The  quantities  of  heat  taken  from  the  sources  are 
Qi,  Qa>  Q*  .  .  .  .     , 


THE    SECOND   LAW   OF   THERMODYNAMICS        63 


the  quantities  of  heat  given  to  the  refrigerators  are 

Q2,  Qa,  Q*  -  -  -  -  , 
and  therefore  the  amounts  of  work  done  by  the  engines  are 

(Qi  -  Qa),  (Q2  -  Qa),  (Qa  -  QO, 

By  Carnot's  theorem  these  depend  only  on  the  temperatures  of 
the  heat  reservoirs,  i.e.,  on  TI,  T2,  T3,  T4,  .  .  .  and  since  the  latter 
are  supposed  to  be  arbitrary,  we  may  arrange  them  so  that  each 
engine  does  exactly  the  same  amount  oj  work  per  cycle  ;  then 
(Qt  -  Qa)  =  (Q2-Q3)  =  (Q3  -  Q4)  =  .  J  .  . 
But  Qa/Qi  =  T2/Ti,     Q3/Q2  =  T3/T2,  .... 


and  so  on. 

Divide  (a)  by  (6)  : 

(Qi  -  Qa)  :  (Qa  -  Qa)  = 
But  (Q!  - 


-  Ta)  :  (Ta  -  T8),  etc. 
=  (Q2  -  Q3),  etc., 


so  that  the  differences  between  the  temperatures  of  the  successive 

heat  reservoirs  are  equal 

when  all  the  members  of 

a     series     of    reversible 

engines,  so   arranged   as 

to     annul     all     thermal 

changes      in      reservoirs 

between  the  first  and  last, 

do     equal     amounts      of 

work. 

The  action  of  this 
series  of  engines  may 
be  represented  on  the 
indicator  diagram  (Fig. 
10)  by  taking  an  iso- 
therm AA',  correspond- 
ing to  TI,  and  crossing  it  by  adiabatics  Aia1}  A2a2,  ....  If 
isotherms  BB',  CC',  ...  are  now  drawn,  corresponding  to 
temperatures  T2,  T3,  .  .  .  so  that  all  the  areas  ABi,  BCi,  ...  are 


FIG.  10. 


64  THERMODYNAMICS 

equal,  the  differences   between  the  temperatures   of   successive 
isotherms  are  also  equal. 

Corollary.—  If  AiA2,  A2A3,  ...  are  portions  of  the  upper 
isotherm,  along  each  of  which  the  same  amount  of  heat  is 
absorbed  as  along  AAi,  show  by  Garnet's  theorem  that  area  ABi  : 
heat  absorbed  along  AAi=  area  AB3  :  heat  absorbed  along  AA3,  and 
thence  that  the  efficiency  of  a  reversible  engine  working  between 
fixed  temperatures  is  independent  of  the  quantity  of  heat 
absorbed  from  the  source. 

The  definition  also  fixes  the  zero  of  absolute  temperature. 
Let  Qi,  Q2  be  the  quantities  of  heat  absorbed  from  the  source 
and  rejected  to  the  refrigerator  at  absolute  temperatures  TI,  T2 
respectively  by  a  reversible  engine.     We  have  proved  that 
Qi  -  Q2  _  T!  -  Ta. 

Qi  T! 

Now  put  T2  =  0, 
.    Qi  -  Q2  _  -. 

~~ 


Hence  the  temperature  of  the  refrigerator  is  zero  when  all  the 
heat  absorbed  from  the  source  is  converted  into  work  by  the 
reversible  engine. 

Corolla  n/  1.  —  Absolute  temperatures  are  essentially  positive 
magnitudes. 

Corollary  2.—  A  body  cooled  to  the  zero  of  absolute  temperature 
("  absolute  zero  ")  cannot  be  made  to  part  with  more  heat.  [Its 
intrinsic  energy  may,  however,  have  any  value,  including  zero,  at 
this  temperature.] 

The  size  of  the  degree  alone  remains  to  be  fixed,  and  is  quite 
arbitrary.  To  produce  as  little  change  as  possible  from  the  ordi- 
nary scale,  Lord  Kelvin  divided  the  range  of  temperature  between 
the  absolute  temperature  of  melting  ice  T0,  and  that  of  boiling 
water,  TI,  into  100  equal  parts,  each  of  which  is  defined  as  one 


By  special  experiments,  to  be  considered  later  on,  he  found 
that  : 

C0  =  T366  very  approximately, 

=  1f  =  0-366, 

o  lo 

• .  TO  =  273°  C.  very  approximately. 


THE    SECOND  LAW  OF   THERMODYNAMICS       65 

The  most  recent  determinations  lead  to  the  value  : 
To  =  273-09°  C. 

Thus,  if  6  is  any  Centigrade  temperature, 
T  =  0  +  273-09. 

In  what  follows,  the  symbol  T  is  always  to  be  understood  as 
referring  to  the  absolute  scale. 

The  boiling-point  of  liquid  helium  is  4'20°  abs.  (Kamerlingh 
Onnes,  Commun.  Pht/s.  Lab.  Leiden,  No.  119,  1911).  A  tempera- 
ture lower  than  1*5°  abs.  has  recently  been  obtained  by  the 
rapid  evaporation  of  solid  helium. 

38.     Analytical  Expression  of  Carnot's  Theorem. 

Theorem. — The  work  obtained  from  a  given  quantity  of  heat 
absorbed  from  the  source  by  a  reversible  engine  is  the  greatest 
amount  which  can  possibly  be  obtained  with  given  temperatures 
of  source  and  refrigerator. 

This  may  be  called  the  maximum  work  for  a  given  quantity  of 
heat  and  a  given  distribution  of  temperatures ;  the  rest  of  the 
heat  is  irrecoverably  lost  so  far  as  its  availability  for  producing 
work  is  concerned,  and  is  therefore  "  wasted,"  but  not  destroyed 
(Lord  Kelvin,  1849). 

Carnot  (loc.  cit.),  who  accepted  (with  forcibly  expressed  doubt) 
the  caloric  theory  of  heat,  according  to  which  heat  is  material 
and  therefore  indestructible,  explained  the  production  of  work 
from  heat  as  brought  about  by  a  "  re-establishment  of  equili- 
brium in  the  caloric,"  the  equilibrium  being  disturbed  by  difference 
of  temperature.  The  motive  effect  of  heat  was  therefore  due  to 
its  "fall  "  from  a  hot  to  a  cold  body,  i.e.,  down  a  precipice  of 
temperature ;  it  was  analogous  to  the  work  done  by  water  falling, 
through  a  water-wheel,  from  a  high  to  a  low  level.  The  above 
theorem,  deduced  by  Carnot  from  this  false  premise  is  how- 
ever strictly  true  when  the  convertibility  of  heat  into  work 
with  a  fixed  rate  of  exchange  is  postulated,  as  was  proved 
by  Clausius  (1850)  and  by  Lord  Kelvin  (1851),  who  at  the  same 
time  pointed  out  that  Carnot's  original  theorem  is  a  limiting  case, 
approached  when  the  temperature  difference  is  infinitesimal.  For : 
Qi  -  Qs  _  TX  -  T2 
Qi  Tx 

.-.  if  we  put  (Ti  —  T2)  =  ST,  and  (Qi  —  Q2)  =  work  done 
=  (SA),  the  brackets  denoting  that  the  enclosed  magnitude 
refers  to  a  cycle,  then 

T.  F 


66  THERMODYNAMICS 

(8 A)  _  ST 

gr:~  TI 

^m 

...  (SA^Q^      .         ....     (a) 

Since  (8A)  is  infinitesimal,  (Qi  —  8A)  =  Q2,  the  heat  discharged 
into  the  refrigerator,  is  very  nearly  equal  to  Qi,  so  that  if  we  put 
Qi  =  Q,  TI  =  T,  and  write  (a) 

(SA)  =  Q  ^     .        '•  -    («') 

we  see  that  the  quantity  of  work  obtained  when  an  amount  of 
heat  Q  passes  by  a  reversible  engine,  from  a  temperature  T  to  a 
temperature  (T  —  BT)  is 

(i.)  proportional  to  the  fall  of  temperature, 

(ii.)  inversely  proportional  to  the  temperature  of  the  source. 

Carnot  assumed  that  (a')  was  true  with  a  finite  temperature 
difference  ;  this,  however,  would  imply  that  no  heat  is  destroyed 
even  in  a  cycle. 

39.    The  Principle  of  Dissipation  of  Energy. 

The  different  forms  of  energy  may  be  classified  according  to 
their  practical  value  as  regards  adaptability,  or  availability,  for  the 
performance  of  useful  work. 

We  assume,  as  a  matter  of  definition,  that  a  raised  weight 
represents  a  distribution  of  energy  most  useful  in  this  respect, 
as  it  merely  requires  to  fall  in  order  to  perform,  on  an  appro- 
priately connected  mechanism,  an  amount  of  work  equal  to  the 
whole  of  the  potential  energy  of  the  system  composed  of  the 
earth  and  weight  in  the  initial  configuration. 

Any  distribution  of  energy  which  can  be  completely  converted 
into  the  potential  energy  of  a  raised  weight  is  called  available 
energy.  If  a  part  only  of  the  given  distribution  of  energy  is 
so  convertible,  we  speak  of  this  as  the  available  part,  and  the 
rest  as  the  unavailable  part,  of  the  total  quantity  of  energy. 

It  is  an  immediate  consequence  of  the  second  law  that  all 
heat  energy  in  a  medium  of  constant  temperature  is  (in  the 
absence  of  all  other  utilieable  distributions  of  temperature) 
completely  unavailable  energy. 

Any  process  whereby  available  energy  is  converted  into 
unavailable  energy  is  called  Dissipation  (or  degradation)  of  Energy. 


THE   SECOND   LAW  OF   THERMODYNAMICS       67 

Examples  of  such  processes  are  the  equalisation  of  tempera- 
ture differences  by  conduction  or  radiation,  the  production  of 
heat  by  friction,  the  expansion  of  gases  into  vacuous  spaces,  and 
the  mixing  of  chemically  different  substances. 

Tln'orem. — A  process  yields  the  maximum  amount  of  available 
energy  irhe-n  it  is  conducted  reversibly. — Proof.  If  the  change  is 
isothermal,  this  is  a  consequence  of  Moutier's  theorem,  for  the 
system  could  be  brought  back  to  the  initial  state  by  a  reversible 
process,  and,  by  the  second  law,  no  work  must  be  obtained  in  the 
whole  cycle.  If  it  is  non-isothermal,  we 
may  suppose  it  to  be  constructed  of  a  very 
large  number  of  very  small  isothermal 
and  adiabatic  processes,  which  may  be 
combined  with  another  corresponding  set 
of  perfectly  reversible  isothermal  and 
adiabatic  processes,  so  that  a  complete 
cycle  is  formed  out  of  a  very  large  number 
of  infinitesimal  Carnot's  cycles  (Fig  11). 
The  work  done  in  such  a  cvcle  is  a  maxi- 


mum when  all  the  operations  are  conducted  Fltt.  u 

reversibly,    and,  since    all    the    auxiliary 

processes   are   reversible,    it    follows    that    the    given    process 

must  also   be  conducted   reversibly   to    obtain   the    maximum 

work. 

It  follows  that  all  irreversible  processes  must  yield  less  than 
the  maximum  amount  of  available  energy,  or  that  all  irreversible 
processes  are  attended  by  dissipation  of  energy. 

The  amount  of  energy  dissipated  in  any  process  is  equal  to  the 
difference  between  the  maximum  available  energy  for  reversible 
execution  and  the  actual  available  energy  for  the  specified 
execution  of  the  process. 

A  distinction  must  be  drawn  between  available  energy  unnecessarily 
dissipitfd  into  unavailable  energy,  by  reason  of  some  irreversibility  inherent 
to  some  part  of  the  process,  and  the  necessary  balance  of  unavailable  energy 
left  in  the  refrigerator  of  a  Carnot's  engine  which  is  working  in  a  perfectly 
reversible  manner. 

The  preceding  considerations  are  summarised  in  a  very  general 
principle,  enunciated  by  Lord  Kelvin  in  1852,  and  called  the 
Principle  of  Dissipation  of  Energy: 

Every  irreversible  process  leads  to  dissipation  of  energy. 

F    2 


68  THERMODYNAMICS 

The  amount  of  energy  dissipated  may  be  taken  as  a  measure 
of  the  amount  of  irreversibility  inherent  in  the  process. 

Further,  since  every  natural  transformation,  or  transference,  of 
energy  is  associated  with  irreversibility  (at  least  in  unorganised 
nature),  it  follows  that  the  whole  store  of  energy  in  that  part  of 
the  universe  where  the  two  laws  of  thermodynamics  hold  good  I'M 
toto,  is  constantly  sinking  lower  and  lower  in  the  scale  of  avail- 
ability. A  little  consideration  shows  that  the  last  stage  must  be 
the  reduction  of  the  whole  of  the  energy  into  heat,  diffused  through 
bodies  at  a  uniform  temperature.  Whether  this  temperature  is 
high  or  low  is  immaterial,  and  all  change,  i.e.,  the  occurrence  of 
phenomena,  must  then  cease,  because  such  energy  is  completely 
unavailable.  It  is  assumed,  of  course,  that  all  chemical  affinities 
are  satisfied,  and  that  all  radio-active  matter  has  decayed  to  its 
ultimate  inactive  stage.1 

40.    Physical  Basis  of  the  Second  Law  of  Thermodynamics. 

Throughout  this  book  the  two  fundamental  laws  of  thermo- 
dynamics are  regarded  as  inductive  generalisations  from  experi- 
ence. They  are  verified  to  a  very  great  degree  of  probability,  no 
single  pertinent  case  among  the  many  thousands  which  have  been 
and  are  being  examined  has  yet  been  found  in  contradiction  to 
them.  In  this  sense  the  laws,  like  all  inductions  from  experience, 
are  empirical,  as  distinguished  from  laws  deduced  by  logical  or 
mathematical  reasoning  from  fundamental  hypotheses.  These 
latter  are  usually  statements  of  the  prevailing  views  on  the 
"  structure  "  of  the  system  concerned,  and  if  the  law  can  be  shown 
to  be  a  consequence  of  the  configuration  and  motion  of  the  parts 
of  a  mechanical  system,  it  is  regarded  as  "  explained."  Thus 
Maxwell,  when  speaking  of  the  deduction  of  Boyle's  law  from  the 
mechanical  kinetic  theory  of  gases,  says  :  "  This  is  Boyle's  law, 
which  is  now  raised  from  the  rank  of  an  experimental  fact  to  that 
of  a  deduction  from  the  kinetic  theory  of  gases."  The  identifica- 
tion of  the  various  forms  of  energy  with  mechanical  energies  of 
hypothetical  systems  is  another  example. 

1  Arrhenius  (Worlds  in  The  Making,  1907)  has  recently  adduced  evidence 
for  the  view  that,  although  dissipation  of  energy  occurs  in  planetary  masses, 
there  may  be  restoration  owing  to  processes  occurring  in  the  nebulse,  and 
that  ' '  the  development  of  the  universe  moves  on  in  a  progressive  cycle,  in 
which  we  can  assume  neither  beginning  nor  end." 


THE   SECOND  LAW  OF  THERMODYNAMICS       69 

The  second  law  as  it  left  the  hands  of  Carnot  required  no  explanation. 
On  the  caloric  theory  then  prevalent,  it  was  a  necessary  consequence  of  a 
hydrodynamical  analogy — the  mechanical  explanation  was  in  fact,  as  Carnot' s 
words  show,  the  source  of  the  principle.  When  the  caloric  theory  was  thrown 
down,  the  analogy  and  explanation  fell  with  it,  and  the  reconstruction  of 
Carnofs  principle  by  Clausius  and  Kelvin  resulted  in  a  law  of  experience. 

A  new  explanation  had  to  be  found,  and  the  fertile  genius  of  Bankine 
(1851)  supplied  it  in  a  peculiar  hypothesis  of  "  molecular  vortices."  This 
representation  of  the  structure  of  material  systems  being  now  obsolete,  it 
is  clear  that  what  was  satisfactorily  explained  to  Bankine  would  now  be 
incomprehensible.  Boltzmann  (1866),  Clausius  (1871),  and  SzQy  (1876),  next 
showed  that  some  special  types  of  dynamical  systems,  involving  so-called 
"stationary  motions,"  could  be  regarded  as  simulating  reversible  thenno- 
dynamic  systems,  provided  the  heat  entering  a  body  was  interpreted  as  the 
increase  of  kinetic  energy  of  its  particles,  and  the  temperature  as  the  mean 
kinetic  energy. 

A  very  exhaustive  investigation  was  carried  out  by  Helmholtz  (1884),  in 
which  an  attempt  was  made  to  interpret  the  second  law,  as  applied  to 
reversible  processes,  on  the  basis  of  the  fundamental  theorem  of  dynamics— 
the  principle  of  Least  Action. 

A  similar  type  of  investigation  is  contained  in  the  work  of  J.  J.  Thomson  : 
"  Applications  of  Dynamics  to  Physics  and  Chemistry,"  where  it  is  >ho\rn 
that,  with  the  ordinary  kinetic  interpretations  of  thermal  magnitudes,  the 
general  equation  of  dynamics  may  without  further  assumptions  be  applied 
to  thermodynamic  systems  and  leads  to  conclusions  in  harmony  with  the 
results  of  pure  thermodynamics. 

Both  Helmholtz  and  Thomson  adopted  Maxwell's  view  that 
irrecersibility  has  its  physical  explanation  in  the  impossibility  of 
controlling  individual  molecular  motions.  Starting  from  the 
kinetic  interpretation  of  temperature,  and  the  conception  of  the 
distribution  of  molecular  velocities  created  by  Boltzmann  and 
himself,  Maxwell  showed  that  the  second  law  is  not  valid  for  some 
particular  systems.  In  these  the  number  of  molecules  is  too 
small  to  form  what  has  been  called  a  "  physically  small "  element 
of  a  material  body,  or  else  the  molecular  motions  are  supposed  to 
be  directly  controllable  by  an  intelligent  being  of  molecular 
dimensions.  The  law  therefore  appears  as  a  statistical  and  not 
as  a  mathematical  truth.  For  if  we  consider  a  mass  of  gas 
enclosed  in  a  vessel,  divided  into  two  parts  by  a  partition  which 
has  a  number  of  small  doors,  and  if  we  station,  at  each  of  these 
doors,  a  being  of  dimensions  so  small  that  he  can  deal  with  the 
individual  molecules,  then  we  can  imagine  a  process  which, 
although  not  violating  the  first  law,  is  in  absolute  contradiction  to 
the  second.  The  temperature  of  the  gas  we  have  to  interpret  as  pro- 


70  THERMODYNAMICS 

portional  to  the  mean  square  velocity  of  the  molecules,  i.e.,  T  a  H*. 
Owing  to  repeated  collisions,  the  velocity  of  any  Delected  mole- 
cule will  fluctuate  within  limits  which,  as  Maxwell's  Distribution 
Law  shows,  are  all  the  less  probable  the  further  they  are  removed 
from  the  value  v^.  If  we  imagine  that  the  beings  stationed 
at  the  doors— called  sorting  demons  by  Maxwell— allow  all  rapidly 
moving  molecules  to  pass  to  one  side  of  the  partition  and  all 
slowly  moving  molecules  to  pass  to  the  other  side,  the  doors  being 
clapped  to  when  unsuitable  molecules  approach,  the  result  would 
be  that  the  gas  on  the  first  side  becomes  hotter,  that  on  the  other 
side  colder.  From  this  arrangement  of  hot  and  cold  reservoirs  a 
finite  amount  of  work  could  be  obtained,  say  by  means  of  a  thermo- 
couple, and  since  this  would  have  been  produced  at  the  expense 
of  the  heat  in  a  body  initially  at  a  uniform  temperature  (i.e.,  such 
that  a  thermometer  shows  no  difference  between  the  temperatures 
of  any  two  parts)  we  have  in  this  arrangement  a  perpetnum 
mobile  of  the  second  class.  The  physical  basis  of  the  second  law, 
which  asserts  that  such  an  arrangement  is  in  contradiction  to 
experience,  is  therefore  to  be  found  in  the  exceedingly  large 
number  of  molecules  of  exceedingly  small  size  which  are  present  in 
a  body  of  finite  size,  and  the  consequent  impossibility  of  controlling 
their  individual  motions  by  any  actual  mechanism.  It  is  as 
though  a  human  being  were  to  attempt  to  direct  the  operations,  at 
every  moment,  of  the  individual  members  of  a  large  colony  of  ants. 
The  production  of  small  temperature  differences  must,  however, 
occur  in  every  gas. 

In  the  rapid  motions  of  small  particles  floating  about  in  a  liquid 
— "  Brownian  movements  " — we  have  an  example  of  motions  pro- 
duced, and  maintained,  in  a  medium  of  uniform  temperature. 
This  is  probably  a  case  in  which  the  simplicity  of  the  system  is, 
comparatively  speaking,  too  great  to  allow  of  the  legitimate 
application  of  the  statistical  method,  which  lies  at  the  basis  of  the 
second  law.  A  mean  value  of  the  kinetic  energy  cannot  be  found. 
It  is  also  quite  an  open  question  whether  the  second  law  is 
applicable  to  living  organisms  ;  the  fineness  of  the  cell-structure, 
and  the  comparatively  enormous — almost  microscopically  visible 
— molecules  of  the  colloidal  substances  occurring  in  the  latter, 
make  it  not  impossible  that  there  are  processes  going  on  there 
which  are  quite  outside  the  consideration  of  thermodynamics. 


THE    SECOND   LAW  OF   THERMODYNAMICS       71 

We  have  already  shown  that  the  first  law  cannot  be  demonstrated 
in  this  case  on  the  basis  of  the  impossibility  of  a  perpetuum 
mobile, 

41.     Entropy. 

In  a  simple  Carnot's  cycle,  in  which  heat  QA  is  absorbed  from 
the  source  at  temperature  TA,  and  heat  QB  is  emitted  to  the 
refrigerator  at  temperature  TB,  we  have  : 

QA  -  QB  _  TA-TB 
QA  TA 

•'•ft-fe  =  ° (1) 

Let  us  now  fix  our  attention  on  the  working  substance,  i.e.,  on 
the  material  system  undergoing  the  cyclic  process.  If  Qi,  Q2  are 
the  quantities  of  heat  absorbed  by  the  system  from  the  source  and 
refrigerator  respectively  : 

Qi  =  QA,  Q2  =  -  QB   •          .          •          •     (2) 

and  if  TI,  T2  are  the  temperatures  of  the  system  when  it  is  absorb- 
ing the  quantities  of  heat  Qi,  Q2  respectively,  the  condition  of 
reversibility  requires  that : 

Ti  =  TA,  Ta  =  TB     ....     (8) 
/.  substituting  from  (2)  and  (3)  in  (1)  we  get 

in  which  every  magnitude  refers  to  the  working  substance,  and 
not  to  the  heat  reservoirs. 

In  the  operations  constituting  a  Carnot's  cycle,  changes  of  Q 
and  T  occur  separately.  In  the  majority  of  cases  both  these  changes 
occur  together,  so  that  the  temperature  of  the  working  sub- 
stance may  be  regarded  as  a  function  of  the  time.  Equation  (4) 
therefore  requires  extension,  and  this  was  effected  by  Lord  Kelvin 
in  May,  1854,  in  the  following  way  : 

If  a  material  system  experiences  a  continuous  action,  or  a 
complete  cycle  of  operations,  of  a  perfectly  reversible  kind,  the 
quantities  of  heat  which  it  takes  in  at  different  temperatures 
are  subject  to  a  homogeneous  linear  equation,  of  which  the 
coefficients  are  the  reciprocals  of  these  temperatures.  If  Q,.  be 


72  THERMODYNAMICS 

the  heat  absorbed  at  temperature  Tr,  this  is  expressed  by  the 
formula  : 


org      =  0        .        .         •         •     (5) 

IVoo/.  Let  there  be  taken,  in  addition  to  this  given  system, 
a  series  of  (71  —  2)  reversible  engines  working  in  the  following 
way:  ^ 

[1]  rejects  heat  Qi  at  Tb  and  absorbs  heat  Qi  rjl2  at  T2  ; 

[2]  rejects  heat  Qi£2  +  Q2  at  T2,  and  absorbs  heat 


[3]  rejects  T3      -1  +         +  Qs  at  T8>  and  absorbs 


[n  -  2]  rejects  Tn_2         +     2  +  .  .  +  +  Q(i_2  at  TH.a 


and  absorbs  T^        +       +  .  .  +     »-=|    at  T..,. 

These  (w  —  2)  auxiliary  engines  constitute  a  material  system 
evolving  heat  Qi  at  TI,  Q2  at  T2,  .  .  QH.2  at  TB.a,  and  absorbing 

heat    Tn  ,  (^  +  ^2  +  •  •  +  sM  at  Tn.r     This  system,  taken 

\ll  12  J-n-2' 

along  with  the  given  one,  constitutes  a  complex  system  causing 
on  the  whole  neither  absorption  nor  emission  of  heat  at  the 
temperatures  TI,  T2,  ...  or  at  any  other  temperatures  than 
TnJl  and  T»,  but  giving  rise  to  an  absorption  or  emission 


at  T«-i,  and  to  an  emission  or  absorption  +  Qn  at  TR. 

This,  having  only  two  temperatures  where  heat  is  absorbed,  is 
subject  to  (4)  ;  hence  : 

Qn  _       i    rT    /Q!     Q2  Qn,.2\          i 

•  p    -    ~  T~  —  7     l,,-i    rrr  Tlf   +  •  •  +  rrr-      +  V»  i 

1  I'  !»-!        L  \ll  J2  A»-"/  J 


THE    SECOND  LAW  OF   THERMODYNAMICS       73 


This  may  be  considered  as  a  general  expression  of  the  Second 
Law,  the  First  Law  taking  the  form 

2A  -  (Q!  +  Q2  +  .  .  Q»)  =  0,  or  2A  -  2Q  =  0, 
where  2  A  is  the  sum  of  the  amounts  of  external  work  performed, 
2Q  the  sum  of  the  amounts  of  heat  absorbed,  in  a  reversible 
cycle. 

If  the  temperatures  of  different  parts  of  the  working  substance 
alter  gradually  during  the  process,  the  sign  of  summation  must 
obviously  be  replaced  by  the  integral  sign,  or  : 

?  =  0          .        .        .        .     (6) 

in  which  BQ  denotes  an  element  of  heat  absorbed  at  tempera- 
ture T  in  any  reversible  cycle,  and  the  integration  is  extended 
round  the  cycle. 

Equation  (6)  was  obtained  in  a  much  less  direct  manner  by 
Clausius  in  December,  1854,  and  is  usually  known  as  the 
Equality  of  Clausius.  It  applies  only  to  reversible  cycles. 

If  the  equality  of  Clausius  is  applied  to  a  reversible  isothermal 
cycle  (T  =  constant)  we  obtain  : 


.'.  (J)  dQ  =  0, 

which  is  one  form  of  Moutier's  theorem. 

Now  consider   any   reversible   change  which  is  not  a  cyclic 
change,  as,  for  example,  the  expansion 
of    a   gas,   or    the    evaporation    of    a 
liquid. 

Theorem.  If  A  and  B  ,are  two 
different  states  of  a  system,  then  the 
value  of  the  integral 

B 

SQ 
T 

is  the  same  for  all  possible  reversible 
processes  in  which  the  state  A  is 
converted  into  the  state  B. 

For  if  AMB,  ANB  are  any  two  such 

reversible   paths  (Fig.  12),   these   taken   together   constitute  a 
reversible  cycle  AMBN,  for  which 


74  THERMODYNAMICS 


where  the  suffixes  denote  the  paths  of  integration. 
••A  /«B 

'«Q\  -  . 
*/* 

by  the  properties  of  definite  integrals  (H.  M.,  §  103), 


which  establishes  the  theorem. 

As  a  particular  case  we  may  instance  the  Caniot's  cycle  ABCD,  Fig.  8. 

»c 

Tr  along  ABC  =  ^, 

Q 

7p-  along  ADC  =  7^, 

'A 

and  these  two  quantities  have  been  shown  to  be  equal. 

/»2 

Thus,  for  reversible  changes,  the  value  of  the  integral      -^ 

•A 

depends  solely  on  the  initial  and  final  states,  and  is  independent 
of  the  path,     -jf   is   therefore   a   perfect   differential   of   some 

function   of  the  variables   defining  the   state    of    the    system 
(H.  M.,  §  115). 

Thus,  in  the  case  of  a  fluid  passing  from  the  state  (p\,  Vi)  to  the  state 
(Pu,  ^2), 

r* 

I    «.r\ 

—  /(Pl>  l'l)    =   <t>  (#2,    ''2)    —    </>  (01,   ^l) 

=  ^  (fla,  P'i)  —  ty  (Oi,  pi). 

This  integral  may  therefore  be  regarded  as  measuring  the 
change  of  some  magnitude  which  depends,  like  the  intrinsic 
energy,  entirely  on  the  actual  state  of  a  material  system,  and  is 
independent  of  the  previous  history  of  the  system.  If  SA,  SB 


THE   SECOND  LAW  OF   THERMODYNAMICS       75 

are  the  values  of  this  magnitude  for  the  states  A  and  B,  we  may 
write  : 


^  =  SB-SA      .         .         .         .     (2) 

A 

the  integral  referring  to  reversible  changes.      S  is  called  the 
entropy. 

Definition. — If  an  element  of  heat,  SQ,  is  added  to  a  system  ly 
a  reversible  process,  at  a  mean  temperature  T,  then  : 

&Q  7Q  /Q\ 

Pp  =  ab  .         .         .         .         .     (o) 

is  called  the  increase  of  the  entropy  of  the  system. 

If  two  states  of  a  system,  A  and  B,  can  be  connected  by  any 

I    £/"v 

reversible  path  of  change,  the  integral        ^  taken  along  this 


f 


path  measures  the  difference  of  the  entropies  of  the  system  in 
the  two  states,  or  : 


...     (4) 

A  JA 

It  follows  that  1/T  is  the  integrating  factor  of  SQ.  Now  since 
SQ  is  a  function  of  two  variables  (in  the  simple  case  of  a  homo- 
geneous fluid),  and  since  the  integrating  factor  of  such  a  magni- 
tude is  usually  also  a  function  of  the  same  two  variables,  we  must 
regard  the  proposition  that  the  integrating  factor  of  SQ  is  a 
function  of  one  variable  only  as  expressing  a  physical,  not  a 
mathematical,  truth. 

Corollary.  In  all  reversible  adiabatic  changes  the  entropy 
remains  constant;  such  changes  are  therefore  isentropic changes. 

It  must  be  emphasised  that  this  holds  good  only  for  reversible 
changes.  To  give  three  instances  of  increase  of  entropy  in 
adiabatic  irreversible  changes  we  may  cite : 

(1)  The  production  of  heat  in  the  system  itself  when  a  mass 
of  viscous  fluid  is  set  in  motion  by  stirring,  and  then  allowed  to 
come  to  rest  by  friction  in  a  vessel  impervious  to  heat. 

(2)  The  mixing,  by  diffusion,  of  two  gases  or  liquids  in  an 
adiabatic  enclosure,  in   wrhich  case  there  is   no   absorption  or 
production  of  heat,  but,  nevertheless,  an  increase  of  entropy. 

(3)  A  chemical  reaction  in  an  adiabatic  enclosure. 
These  cases  will  be  taken  up  later. 


76  THERMODYNAMICS 

42.     Specification  of  Entropy. 

Just  as  the  intrinsic  energy  of  a  body  is  defined  only  up  to  an 
arbitrary  constant,  so  also  the  entropy  of  the  body  cannot,  from 
the  considerations  of  pure  thermodynamics,  be  specified  in  abso- 
lute amount.  We  therefore  select  any  convenient  arbitrary  stan- 
dard state  a,  in  which  the  entropy  is  taken  as  zero,  and  estimate 
the  entropy  in  another  state  /3  as  follows  :  The  change  of  entropy 
being  the  same  along  all  reversible  paths  linking  the  states  a  and 
y8,  and  equal  to  the  difference  of  the  entropies  of  the  two  states, 
we  may  imagine  the  process  conducted  in  the  following  two 
steps : 

(i.)  Take  the  body  from  a  to  y  along  the  adiabatic  containing 
a,  where  y  is  the  intersection  of  this  with  the  isotherm  containing 
/3.  The  change  of  entropy  is  zero. 

(ii.)  Take  the  body  from  y  to  /3  along  the  isotherm.  If  Q^  is 
the  heat  absorbed  and  T^  the  constant  temperature,  the  entropy 

in  the  state  3  is  ^.     This  divided  by  the  mass  of  the  body  is 

the   entropy  per    unit   mass,  which  we    shall  call  the  specific 
entropy. 

If  there  are  a  number  of  separate  masses,  mi,  m2,  .  .  ?nif  the 
total  entropy  of  the  system  is  equal  to  the  sum  of  the  entropies 
of  the  i  separate  masses  plus  the  entropy  S,M  of  any  medium  in 
which  they  are  contained  : 


If  the  masses  are  homogeneous,  and  *b  *a, .  .  *f  are  their  specific 
entropies : 

Si  i  I       O 

—  Wll^l  -j-  7??2^2     i     •     •  '"j^i  T   O    . 

Any  entropy  which  the  system  may  possess  in  virtue  of  the  mutual 
actions  of  the  masses  is  taken  as  included  in  Sm. 
A  similar  expression  holds  for  the  intrinsic  energy : 
U  =  niiii i  -j-  w2»2  -f-  .  .  -j-  m-iUi  -\-  Um. 

43.     The  Entropy-Temperature  Diagram. 

If  we  take  rectangular  axes  and  put  x  ~  S,  y=  T,  the  Carnot's 
cycle  will  be  represented  by  a  rectangle  ABCD,  consisting  of  two 
isotherms  BC,  DA  and  two  isentropics  (adiabatics)  AB  CD 
(Fig.  13). 


THE   SECOND  LAW  OF   THERMODYNAMICS        77 

The  heat  absorbed  in  the  cycle 

=  the  heat  absorbed  along  BC— heat  rejected  along  DA 

=  (Ta-TO(S8-SO 

=  area  of  cycle. 

The  area  is  positive  if  traced  out 
clockwise.  Since  the  heat  absorbed 
in  the  cycle  is  equal  to  the  work 
done,  the  areas  of  the  Carnot's  cycle 
on  the  (p,  v)  and  (S,  T)  diagrams 
are  equal.  This  may  be  generalised 
to  apply  to  any  reversible  cycle 

where  the  only  external  work  is  done     

by  expansion. 

For  a  small  reversible  change : 

dU  =  6Q  -  8A  =  TtfS  - 


FIG.  13. 


Corollary  1.  The  area  included  in  the  pv  plane  between  any 
two  adiabatics  (Si,  82)  and  any  two  isotherms  (Ti,  T2)  is  equal  to 
(T2  —  TI)  (S2  —  Si),  and  this  represents  the  heat  absorbed  in  the 
cycle. 

Corollary  2.  If  we  could  draw  on  the  j>r  plane  the  isothermal 
line  of  absolute  zero  (T  =  0)  the  area  included  between  it,  any 
two  adiabatics,  and  an  isotherm  T  would  represent  the  heat 
absorbed  in  passing  along  the  upper  isotherm  from  one  adiabatic 
to  the  other. 

Corollary  3.  If  any  path  of  reversible  change  is  drawn  on  the 
pi-  plane  between  two  adiabatics,  the  area  between  it  and  the 
absolute  zero  isotherm  represents  the  heat  absorbed  in  the 
change  (Zeuner). 


44.     Entropy  and  Unavailable  Energy. 

If  a  quantity  of  heat  Qi  is  taken  from  a  body  at  temperature 
the  maximum  amount  of  work  obtainable  from  it  is  : 


-  Q2  =  Ql     l  - 


78  THERMODYNAMICS 

whilst  the  balance  : 


is  wholly  unavailable  energy  given  up  as  heat  to  the  refrigerator 
at  the  lowest  temperature  T2. 

This  follows  from  the  result  established  for  the  Carnot's  cycle  : 

QL_O? 
T!  ~  T2' 

and  the  proposition  that  it  is  impossible  to  get  more  work  from  a 
given  quantity  of  heat  than  we  can  get  from  it  in  a  Carnot's  cycle. 
The  maximum  amount  of  work  obtainable  from  a  given 
quantity  of  heat,  called  its  motivity  by  Lord  Kelvin  (1852),  is 
thus  always  less  than  the  mechanical  equivalent  of  the  quantity 
of  heat,  except  in  the  limiting  case  when  the  refrigerator  is  at 
absolute  zero  (T2  =  0).  It  cannot  be  specified  in  terms  of  the 
condition  of  the  body  from  which  the  heat  is  taken,  or  into  which 
the  heat  passes,  but  requires  in  addition  a  knowledge  of  the 
lowest  available  temperature,  T2.  For  if  we  had  another  body  at 
temperature  T0,  where  T0  <  T2,  which  could  be  used  as  a 
refrigerator,  the  amount  of  work  : 

^-°\        r\    /^2        T^-° 

T2J  -~~  Ql  ITT  -  Ti 

could  be  obtained  further  from  Q2,  and  the  unavailable  energy  : 

Qo  =  Q2  X  £-°  =  Qi  X  £°  .         .         .         (4) 

la  li 

would  go  to  the  refrigerator,  the  final  result  being  the  same  as 
if  the  cycle  had  been  performed  directly  between  the  temperatures 
TI  and  T0. 

Equations  (2)  and  (4)  show  that  the  unavailable  part  of  Q  is 
directly  proportional  to  the  lowest  available  temperature  and 
inversely  proportional  to  the  temperature  of  the  body  from  which 
Q  is  taken.  Again,  since 

Qo  _  Qi  _  Qi 
To  -  T2  ~  IV 

we  see  that  Q/T  is  independent  of  the  lowest  available  tempera- 
ture, and  from  (2)  and  (4)  that  Q/T  only  requires  to  be  multiplied 
by  the  lowest  available  temperature  to  give  the  unavailable  part, 

T 
Q  X  r-p  of  the  heat  taken  from  a  body  at  temperature  T. 

But  Q/T  is  the  entropy  of  the  quantity  of  heat  Q  at  T  ;  hence  : 


THE    SECOND   LAW  OF   THERMODYNAMICS       79 

unavailable  energy  =  entropy  X  lowest  available  temperature,  which 
is  an  alternative  definition  of  entropy. 

45.     Inequality   of  Clausius. 

If  a  body  absorbs  an  amount  of  heat  Q  from  a  reservoir  at 
temperature  T,  and  at  the  same  time  does  work  A, 

/  T  \ 

the  gain  of  available  energy  =  Q  M  —  ^J  , 

the  loss  of  available  energy  =  A, 

To  being  the  lowest  available  temperature, 
.  '  .  the  nett  gain  of  available  energy  (*)  is 


=  AU  -  Q°, 

this  being  the  amount  by  which  the  capacity  of  the  body  for 
doing  work  is  increased. 

We  have  assumed  that  the  temperatures  remain  constant 
during  the  transference  of  a  finite  amount  of  heat  Q,  which 
implies  that  the  heat  reservoirs  have  very  large  heat  capacities. 
To  remove  this  restriction,  we  suppose  that  the  amount  of  heat 
absorbed  is  infinitesimal,  6Q.  Then,  for  the  gain  of  available 
energy  we  have  : 


=  rfU  -  26Q  TJ?, 

where  -5Q,  2SA  are  the  sums  of  the  amounts  of  heat  absorbed 
from,  and  external  work  done  upon,  all  the  bodies  outside  the 
system,  and 

rfU  =  25Q  -  2SA 
is  the  increase  of  intrinsic  energy  of  the  system. 

The  increase  of  available  energy  in  a  finite  change  is  therefore  : 


=    I    «T  -  2  j    8Q  ^ 
r       r       TS  f28Q 

=    1-2   —     Ul  —     J-O^     I       -jjf 

J\ 


.  (1) 


80  THERMODYNAMICS 

If  the  changes  constitute  a  closed  cycle : 
U2  -  Ui  =  0 

>'$  (2) 

Now  if  any  irreversible  changes  occur  in  the  system  itself 
during  the  execution  of  the  cycle,  the  principle  of  dissipation  of 
energy  shows  that  the  available  energy  will  be  diminished  in 
virtue  of  these,  and  since  the  available  energy  of  the  system  must 
be  the  same  after  as  it  was  before  the  execution  of  the  cycle, 
because  the  state  of  the  system  is  unaltered,  it  follows  that  some 
available  energy  must  have  been  absorbed  from  outside  in  con- 
nection with  the  absorption  of  heat;  hence  (A*)  and  therefore  also 

r  $\r\  r  ^o 

—  T02(   )-^must  be  positive,  and  hence  (])-TJT  must  be  negative, 

or  v/MQ^n 

P  ~^  "         '         •     w) 

This  is  called  the  Inequality  of  Clausius,  who,  however,  estab- 
lished it  in  a  different  way. 

If  the  cycle  is  reversible,  there  is  no  dissipation  of  energy,  and 

.     (Sa) 

which  is  the  result  established  in  §  41. 

The  integral  of  (3)  must  be  interpreted  as  follows :  T  refers  to  the 
temperature  of  the  body  from  which  the  element  of  heat  8Q  is  taken, 
and  the  integral  sums  up  all  the  quantities  SQ/T  for  that  body. 
The  symbol  2  further  extends  this  to  all  the  external  bodies  con- 
cerned. Thence  the  sum  of  all  the  magnitudes  SQ/T  is  negative. 
Now  SQ/T  represents  the  entropy  lost  by  the  external  body  during 
the  small  change,  because  SQ,  being  the  heat  absorbed  by  the 
system,  will  be  heat  lost  by  the  external  body,  and  the  relations 
(3)  and  (3a)  may  therefore  be  expressed  in  words  as  follows : 

If  any  cyclic  process  is  performed  with  a  given  material  system, 
the  entropy  of  all  the  surrounding  bodies  which  have  in  any  way 
been  involved  in  the  process,  either  as  emitters  or  absorbers  of 
heat,  either  remains  unchanged,  if  the  cycle  is  reversible,  or  else 
increases,  if  the  cycle  is  performed  irreversibly. 

Now  let  us  consider  any  process  which  is  not  a  cyclic  process, 
and  in  which  the  system  is  taken  from  an  initial  state  [1]  to  a 
final  state  [2].  We  shall  prove  that  if  Si,  S2  are  the  entropies  of 


THE    SECOND  LAW  OF   THERMODYNAMICS        81 

the  system  in  its  initial  and  final  states,  and  -^  has  the  signifi- 
cance of  the  preceding  paragraph,  then 

/*2 
5     ^<S,-B:    .  .         .      (4) 


according  as  the  change  is  not,  or  is,  conducted  reversibly. 

We  assume  that  there  is  conceivably  some  reversible  process,  a, 
by  means  of  which  the  state  [2]  can  be  reconverted  into  the  initial 
state  [1],  after  the  execution  of  the  given  process  3.  The  processes 
a  and  ft  constitute  a  cycle,  and 


according  as  ft  is  not,  or  is,  reversible, 
But 


since  a  is  reversible,  hence  if  ft  is  reversible, 

V    I      y          Q  Q  I f^\ 

2    I   -    --    =  b2  —    Oi        .  .  .  .       (3) 

J  l 
/3rev. 

but  if  ft  is  not  reversible, 


fe 


^im 

/•a 

I    STk 

^*  Q  Q  / R\ 


pin. 

Relations  (5)  and  (6)  are  combined  in  (4).  Inequality  (6) 
shows  that  if  a  system  undergoes  any  irreversible  change,  there 
will  have  been  a  positive  amount  of  entropy  generated  in  the 
system  itself  during  that  change,  since  the  entropy  absorbed  from 
outside  is  always  less  than  the  amount  by  which  the  entropy  of 


the  system  increases.     This  holds  good  even  if  2-     =  0,   i.e., 
if  no  entropy  at  all  is  taken  in  from  outside. 


82  THEKMODYNAMICS 

We  shall  now  suppose  the  change  to  be  very  small,  so  that  (6) 
can  be  written  : 


orSQ<TWS        ....     (7) 

This  involves  no  loss  of  generality,  because  all  the  results  could 
equally  well  have  been  obtained  from  (6),  which  refers  to  a  finite 
change,  but  the  form  adopted  is  more  convenient  in  application. 

Inequality  (7)  shows  that  the  heat  absorbed  from  outside  is 
less  than  corresponds  with  the  increase  of  entropy  of  the  system 
(viz.,  8Q  =  Tt/S,  where  SQ  is  the  heat  absorbed  in  the  process 
when  conducted  reversibly),  hence  some  entropy  is  generated 
inside  the  system  itself. 

Now       -  2SQ  =  <?U  +  SSA, 

where  SA  is  an  element  of  work  done  in  the  irreversible  change, 
hence  : 


if  the  process  is  irreversible,  whilst 

TdS  =  dU  +  2dA 
if  the  process  is  reversible. 

The  two  expresions  can  be  combined  into  : 

TtZS  ?!  dll  +  2SA  .  .  .  .  (8) 
the  inequality  referring  to  irreversible,  and  the  equality  to 
reversible,  processes. 

Now  let  us  suppose  that  the  change  is  adiabatic,  so  that  there 
is  no  transfer  of  heat,  but  possibly  the  performance  of  work,  then 

28Q  =  f/U  -|-  2SA  =  0 

/.  TdS  ~  0,  or 

(SS)Q>0   .        .        .        .        .     (9) 

so  that  in  irreversible  processes  which  are  adiabatic  the  entropy 
increases  whilst  the  energy  may  either  increase  or  diminish, 
according  to  the  sign  of  28  i.  This  explains  the  statement  of 
§  41  that  an  adiabatic  change  is  necessarily  an  isentropic  change 
only  when  it  is  reversible,  for  then  the  lower  sign  is  taken  in  (9) 
and 

(8S)Q  =0. 

Further,  let   us   suppose  the  system  completely  isolated,  by 
enclosing  it  in  a  vessel  with  perfectly  rigid   walls,  which  are 


THE   SECOND  LAW  OF  THERMODYNAMICS       83 

perfect  non-conductors  of  heat.      Then   28A  =  2SQ  =  dU  =0, 
so  that 


or  (SS^  >0  ....  (10) 
In  this  case  there  is  an  increase  of  entropy  in  an  irreversible 
process,  whilst  the  energy  remains  constant.  This  result  brings 
out  clearly  the  independence  of  the  two  fundamental  principles 
of  thermodynamics,  the  first  law  dealing  with  the  energy  of  a 
system  of  bodies,  and  the  second  law  with  the  entropy. 

46.  The  Aphorism  of  Clausius. 

If  the  system  is  not  isolated,  its  entropy  may  either  increase 
or  decrease.  Thus,  if  a  mass  of  gas  is  compressed  in  a  cylinder 
impervious  to  heat,  its  entropy  increases,  but  if  heat  is  allowed  to 
pass  out  into  a  medium,  the  entropy  of  the  gas  may  decrease.  By 
including  the  gas  and  medium  in  a  larger  isolated  system,  we 
can  apply  (10)  of  §  45,  and  hence  show  that  the  medium  gains 
more  entropy  than  the  gas  loses.  An  extended  assimilation  of  this 
kind  shows  that,  if  every  body  affected  in  a  change  is  taken  into 
account,  the  entropy  of  the  whole  must  increase  by  reason  of 
irreversible  changes  occurring  in  it.  This  is  evidently  what 
Clausius  (1854)  had  in  mind  in  the  formulation  of  his  famous 
aphorism  :  "  The  entropy  of  the  universe  strives  towards  a 
maximum."  The  word  "  universe  "  is  to  be  understood  in  the 
sense  of  an  ultimately  isolated  system. 

47.  Compensating  Changes. 

In  a  Carnot's  cycle,  the  entropy  Qi/Ti  is  taken  from  the  hot 
reservoir,  and  the  entropy  Q2/T2  is  given  up  to  the  cold  reservoir, 
and  no  other  entropy  change  occurs  anywhere  else.  Since  these 
two  quantities  of  entropy  are  equal  and  opposite,  the  entropy 
change  in  the  hot  reservoir  is  exactly  balanced,  or,  to  use  an 
expression  of  Clausius,  is  compensated  by  an  equivalent  change  in 
the  cold  reservoir.  Again,  in  any  reversible  cycle  there  is  on 
the  whole  no  production  of  entropy  so  that  all  the  changes  are 
compensated. 

If  now  we  have  any  reversible  change  which  is  not  a  cycle, 
there  will  be  a  change  of  entropy  in  the  system,  but  this  will 
have  a  compensating  change  outside  the  system.  For  suppose 


84  THERMODYNAMICS 

the  entropy  of  the  system  decreases,  then  the  entropy  of  the 
external  bodies  might  increase,  decrease,  or  remain  constant.  It 
cannot,  however,  either  decrease  or  remain  constant,  for  then 
we  should  have  the  entropy  of  the  whole  system  decreasing, 
which  is  impossible.  It  must  therefore  increase,  and  this  increase 
must  be  exactly  equal  to  the  decrease  in  the  given  system ;  for 
in  virtue  of  the  assumed  reversibility  of  the  change,  we  could 
reverse  the  sign  of  every  entropy  change  by  taking  the  system 
back  to  its  initial  state,  and  there  would  be  a  decrease  of  entropy 
if  the  external  increase  in  the  first  case  had  been  greater  than 
the  decrease  in  the  system.  The  change  is  therefore  compen- 
sated. Again,  if  we  had  supposed  the  entropy  of  the  system  to 
increase,  we  should  simply  have  to  reverse  all  the  processes  to 
arrive  at  the  first  case,  and  hence  the  process  is  compensated. 
Thus  every  reversible  process  admits  of  a  compensating  process. 

But  if  the  given  process  is  conducted  irreversibly,  we  have 
proved  that  there  is  always  more  entropy  generated  in  the  system 
than  is  Lost  by  bodies  outside  the  system,  and  the  excess  is  called 
the  non-compensated  entropy.  It  may  happen,  and  frequently 
does,  that  the  entropy  of  the  system  itself  decreases  in  a  par- 
ticular change,  but  we  have  proved  that  there  is  an  increase 
outside  the  system  which  is  greater  than  the  decrease  in  the 
system,  or  at  best  equal  to  it  in  the  case  of  reversible  changes. 

The  application  of  the  principle  of  entropy  to  irreversible  processes  has 
given  rise  to  much  discussion  and  controversy.  The  exposition  here 
adopted  is  based  on  the  investigations  of  Lord  Kelvin  (1852)  in  connexion 
with  Dissipation  of  Energy. 

48.     Examples  of  Irreversible  Changes. 

We  shall  conclude  this  chapter  by  considering  a  few  typical 
cases  of  systems  undergoing  changes  attended  by  intrinsic  or 
conditional  irreversibility. 

(1)  Conduction  of  Heat. — Let  the  quantity  of  the  heat  8Q  pass 
by  conduction  from  a  body  at  temperature  Tx  to  a  body  at  tem- 
perature T2.  Suppose  that  an  auxiliary  medium  at  temperature 
TO  is  available. 

m     rp 

[  .  The  motivity  of  SQ  at  1\  is  SQ  .  — ^ — °,  and  its  motivity  at 

m     •      $n      Ta  —  TO 

12  is  6(J  . pp . 

-La  .         - 


THE    SECOND   LAW  OF   THERMODYNAMICS       85 

The  loss  of  motivity  is  therefore 

Tl  -  T°      Ta  ~  T°1  -  T  SO  (  l         M 

-~     --   -:  T°8Q      -      • 


Since  this  is  positive,  by  the  principle  of  dissipation  of  energy, 
1/T,  >  1/Ti  .'.  T!  >  Tfc 

so  that  heat  passes  by  conduction  from  a  hotter  to  a  colder  body. 

The  loss  of  entropy  of  the  hotter  body  is  7^,  and  the  gain  of 
entropy  of  the  colder  body  is  ~, 

±'2 

.'.the  increase  of  entropy  of  the  whole  system  consequent  upon 
the  occurrence  of  the  irreversible  change  is 


From  what  precedes  we  see  that  : 

(1)  The  gain  of  entropy  is  positive. 

(2)  The  gain  of  entropy  is  equal  to  the  dissipated  energy  pro- 
duced (or  the  available  energy  lost)  divided  by  the  temperature 
of  the  auxiliary  medium.     It  is  easy  to  generalise  this  result  for 
all  processes. 

Loss  of  motivity  (dissipation  of  energy)  is  therefore  accom- 
panied by  increase  of  entropy,  but  the  two  changes  are  not  wholly 
co-extensive,  because  the  former  is  less  the  lower  the  tempera- 
ture TO  of  the  auxiliary  medium,  whilst  the  latter  is  independent 
of  TO,  and  depends  only  on  the  temperature  of  the  parts  of  the 
system.  If  T0  =  0,  i.e.,  the  temperature  of  the  surroundings 
is  absolute  zero,  there  is  no  loss  of  motivity,  whilst  the  entropy 
goes  on  increasing  without  limit  as  the  heat  is  gradually 
conducted  to  colder  bodies. 

Similar  considerations  apply  to  passage  of  heat  from  one  body 
to  another  by  radiation.  In  this  case  the  energy,  in  its  transition 
from  one  body  to  the  other,  exists  as  radiant  energy  in  the  ether. 
We  have  therefore  to  suppose,  when  the  energy  leaves  the  hot 
body  and  so  reduces  its  entropy,  that  it  must  carry  entropy  into 
the  ether. 

(2)  Expansion  of  a  Gas  into  a  Vacuum.  —  If  a  gas  is  allowed  to 
rush  into  a  vacuous  space,  or  into  a  space  containing  a  gas  under 
a  less  pressure,  we  have  an  example  of  a  process  attended  by 
conditional  irreversibility. 

Let  a  volume  i\  of  an  ideal  gas  be  put  into  communication  with 


86  THERMODYNAMICS 

a  vacuous  vessel  of  volume  r2.  It  rushes  into  the  latter,  and 
occupies  a  volume  ri  -f-  r2.  No  work  has  been  done,  hence  the 
energy  dissipated  is  equal  to  the  work  which  could  have  been 
obtained  had  the  expansion  been  performed  isothermally  and 
reversibly. 

The  gas  may  be  restored  to  its  initial  state  by  compressing, 
and  removing  the  heat  generated  by  conducting  it  away  to  the 
medium.  Since  it  is  an  experimental  fact  that  no  heat  is  emitted 
or  absorbed  in  the  process  of  free  expansion,  this  heat  is  the 
exact  equivalent  of  the  work  spent  in  the  reversible  compression, 
since  no  energy  change  occurred  during  the  previous  expansion. 
There  is  no  change  of  quantity  of  the  energy  in  allowing  a 
gas  to  expand  irreversibly,  and  then  bringing  it  back  reversibly 
to  its  initial  state.  There  is,  however,  a  change  in  the  quality  of 
the  energy,  because  from  a  quantity  of  useful  work  we  obtain  an 
equivalent  of  useless  heat  in  a  reservoir  at  a  uniform  tempera- 
ture— in  other  words,  there  has  been  a  dissipation  of  energy. 
We  could  imagine  the  process  reversed  without  dissipation  if  we 
supposed  the  expanded  gas  enclosed  in  a  cylinder  closed  by  a 
piston  which  is  pushed  in  bit  by  bit  by  an  army  of  Maxwell's 
demons,  each  element  of  the  piston  being  advanced  when  free 
from,  and  kept  rigid  when  exposed  to,  molecular  bombardment. 
Thus  the  availability  lost  in  the  expansion  could  be  restored 
without  any  compensating  dissipation.  Such  a  restoration  would 
naturally  contradict  both  the  principle  of  dissipation  of  energy 
and  the  principle  of  increasing  entropy,  and  their  basis  the 
Second  Law  of  Thermodynamics.  This  violation  is,  however, 
purely  imaginary,  because  Maxwell's  demons  do  not  exist. 

(3)  Mia-iny  of  Gases  by  Dl/mion. — Exactly  similar  considera- 
tions apply  to  the  spontaneous  intermingling  of  two  gases  by  diffu- 
sion, the  increase  of  entropy  being  calculable  from  the  isothermal 
absorption  of  heat  when  the  process  is  carried  out  reversibly  by 
means  of  semi-permeable  septa,  as  described  in  §  123.     The  pro- 
cess could  be  reversed  in .  imagination,  and  the  lost  availability 
restored,  by  demons  which  would  allow  molecule  s  of  one  gas  to 
pass  in  one  direction,  those  of  the  other  gas  in  the  opposite 
direction,  across  a  partition  dividing  the  mixture  into  two  parts 
equal  respectively  to  the  initial  volumes  of  the  unmixed  gases, 
but,  of  course,  such  a  process  is  physically  impossible. 

(4)  Collision. — If  a  mass  moving  in  any  direction  is  suddenly 


THE    SECOND   LAW  OF   THERMODYNAMICS       87 

arrested  in  its  course  by  striking  against  a  non-conducting  wall, 
the  kinetic  energy  is  converted  into  heat  in  the  body,  and  this 
implies  a  corresponding  increase  of  entropy  o!  the  latter.  A 
similar  result  follows  if  two  or  more  imperfectly  elastic  spheres 
come  into  collision,  and  if  we  are  given  a  system  of  such  spheres 
in  motion,  they  must  ultimately,  by  mutual  collision,  be  brought 
to  rest,  their  kinetic  energy  being  converted  into  heat. 

(5)  Viscosity. — If  a  mass  of  viscous  liquid  is  set  in  rotation  in 
a  rough  vessel,  the  kinetic  energy  is  gradually  dissipated  by 
friction,  and  the  liquid  comes  to  rest  in  a  slightly  warmed  con- 
dition. The  increase  of  entropy  is  in  this  case  due  to  heat 
generated  in  the  system  itself. 

Among  the  causes  producing  irreversibility  we  may  instance 
the  forces  depending  on  friction  in  solids,  viscosity  of  liquids  ; 
imperfect  elasticity  of  solids  ;  inequalities  of  temperature  (leading 
to  heat  conduction)  set  up  by  stresses  in  solids  and  fluids  ; 
generation  of  heat  by  electric  currents ;  diffusion  ;  chemical  and 
radio-active  changes  ;  and  absorption  of  radiant  energy. 

The  presence  of  any  type  of  irreversibility  inevitably  leads  to 
dissipation  of  energy,  and  therefore  to  increase  of  entropy. 

The  physical  (or,  rather,  the  mechanical)  interpretation  of 
entropy  is  identical  with  the  problem  of  the  interpretation  of  the 
Second  Law  of  Thermodynamics,  and  the  attempts  at  its  solution 
by  Boltzmann,  Clausius,  etc.,  have  already  been  referred  to.  In 
this  connexion  the  treatment  of  irreversible  processes  offers  con- 
siderable difficulty.  The  first  step  in  this  direction  was  Lord 
Kelvin's  "  Kinetic  Theory  of  Dissipation  of  Energy"  (1874),  in 
which  the  relation  between  irreversibility  and  the  technical 
impossibility  of  getting  to  grips  with  the  individual  molecules 
and  controlling  their  motions  was  explained.  In  this  we  find 
the  first  application  of  the  theory  of  probabilities  to  problems  of 
the  kind  contemplated.  As  Kelvin  observes :  "  If,  then,  the 
motion  of  every  particle  of  matter  in  the  universe  were  precisely 
reversed  at  any  instant,  the  course  of  nature  would  be  simply 
reversed  for  ever  after.  The  bursting  bubble  of  foam  at  the  foot 
of  a  waterfall  would  reunite  and  descend  into  the  water ;  the 
thermal  motions  would  reconcentrate  their  energy,  and  throw 
the  mass  up  the  fall  in  drops  re-forming  into  a  close  column  of 
ascending  water.  Heat  which  had  been  generated  by  the  friction 
of  solids  and  dissipated  by  conduction,  and  radiation  with 


88  THERMODYNAMICS 

absorption,  would  come  again  to  the  place  of  contact,  and  throw 
the  moving  body  back  against  the  force  to  which  it  had  previously 
yielded.  Boulders  would  recover  from  the  mud  the  materials 
required  to  rebuild  them  into  their  previous  jagged  forms,  and 
would  become  reunited  to  the  mountain  peak  from  which  they 
had  formerly  broken  away.  And  if  also  the  materialistic  hypo- 
thesis of  life  were  true,  living  creatures  would  grow  backwards, 
with  conscious  knowledge  of  the  future,  but  no  memory  of  the 
past,  and  would  become  again  unborn.  But  the  real  phenomena 
of  life  infinitely  transcend  human  science;  and  speculation 
regarding  consequences  of  their  imagined  reversal  is  utterly 
unprofitable." 

If  every  natural  process  could  be  represented  by  a  dynamical 
equation,  the  substitution  of  —  t  for  t  in  the  equation  (t  being 
time)  would  lead  to  an  equation  describing  the  exactly  reversed 
process.  If  we  could  represent  the  actual  process  on  a  cinemato- 
graph film,  the  reversed  process  would  be  seen  when  the  film 
was  put  backwards  through  the  machine,  and  events  like  those 
just  described  would  unfold  themselves  to  our  view.  That  such 
phenomena  do  not  appear  in  nature  is  a  consequence  of  the 
irreversibility  of  every  process. 

A  great  advance  was  made  in  the  direction  of  the  physical  inter- 
pretation of  entropy,  and  in  the  systernatisation  of  irreversible  pro- 
cesses, when  L.  Boltzmann  (1877)  showed  that  the  definition  of 
the  entropy  could  be  regarded  as  a  problem  in  the  theory  of  proba- 
bilities ("  Ueber  die  Beziehung  zwischen  dem  zweiten  Hauptsatz 
der  mechanischen  Warmetheorie  und  der  Wahrscheinlich- 
keitsrechnung  respektive  den  Sa'tzen  iiber  das  Warmegleich- 
gewicht,"  Wiss.  AbltL,  II.,  164).  The  second  law,  and  the  pheno- 
mena of  irreversibility,  depend  not  on  any  peculiarity  of  the 
motions  constituting  heat  themselves,  but  on  the  statistical  pro- 
perties of  systems  with  an  enormous  number  of  degrees  of  free- 
dom, which  are  realised  in  bodies  composed  of  an  exceedingly 
large  number  of  atoms.  Whereas  to  an  army  of  Maxwell's  demons 
the  "  state  "  of  a  gas  would  be  completely  defined  by  the  aggregate 
of  the  position  and  velocity  co-ordinates  of  each  separate  mole- 
cule, i.e.,  by  6»  variables,  when  there  are  »  molecules,  and  every 
change  of  state,  being  a  change  in  a  dynamical  system,  would  be 
completely  reversible,  yet  to  an  ordinary  observer  the  state  would 
be  defined  by  two  variables,  the  temperature  and  density.  Both 


THE    SECOND  LAW  OF   THERMODYNAMICS        89 

of  these,  however,  have  only  a  statistical  significance  (§  20). 
The  conception  of  entropy  as  measuring  the  probability  of  a 
state  has  been  extended  by  Planck  to  radiation,  and  no  doubt 
will  lie  at  the  basis  of  future  developments  of  thermodynamics. 
(Cf.  Planck,  Theorie  dcr  Warmestrahlung,  Leipzig,  1906,  pp.  129 
ct  seq.) 


CHAPTER  IV 

THE    THERMODYNAMIC    FUNCTIONS   AND    EQUILIBRIUM 

49.     Equilibrium  and  Stability. 

A  system  is  in  equilibrium  when  the  position  of  each  of  its 
parts,  and  the  state  of  that  part,  remain  unchanged  with  lapse 
of  time. 

If  any  one  of  the  conditions  maintaining  the  system  in  its 
equilibrium  state  (r.(j.  temperature  or  pressure)  is  changed,  the 
state  of  the  system  is  also  changed  and  the  equilibrium  is  dis- 
placed. If  the  change  of  external  conditions  is  very  small,  the 
displacement  of  the  equilibrium  may  also  be  correspondingly 
small,  and  may  be  reversed  when  the  displacing  change  is 
reversed.  The  state  of  equilibrium  is  then  called  a  state  of  true 
equilibrium.  A  dynamical  illustration  is  afforded  by  a  spring 
extended  by  a  weight.  If  the  weight  is  changed  by  an  infinitesi- 
mal amount,  there  is  a  correspondingly  small  alteration  of  the 
length  of  the  spring,  and  if  the  original  weight  is  restored,  so 
also  is  the  original  length.  A  physical  illustration  is  afforded  by 
a  mixture  of  water  and  steam  in  a  cylinder  at  100°  C.  under  an 
external  pressure  of  1  atm.,  and  subjected  to  small  evaporative 
or  condensative  changes  consequent  on  small  changes  of  tem- 
perature and  pressure. 

If  the  spring  and  weight  had  been  immersed  in  oil  or  treacle, 
a  viscous  reaction  would  have  been  opposed  to  the  motion,  but 
since  this  is  proportional  to  the  velocity,  it  can  be  made  as  small 
as  we  please  by  executing  the  process  very  slowly  and  vanishes 
in  the  limit.  Viscous  reactions  do  not,  therefore,  prevent  a 
system  from  existing  in  a  state  of  true  equilibrium  provided  all 
changes  are  made  infinitely  slowly ;  they  merely  retard  change, 
and  the  retardation  vanishes  in  the  limit. 

A  state  of  equilibrium  which  does  not  satisfy  the  conditions 
for  true  equilibrium  is  called  a  state  of  false  equilibrium.  A 
system  may  remain  in  a  given  state  for  a  long  period  of  time, 
and  thus  appear  to  be  in  an  equilibrium  state.  A  small  change 


THEEMODYNAMIC   FUNCTIONS  AND  EQUILIBRIUM     91 

in  the  external  conditions  produces,  however,  either  no  change 
at  all,  or  else  a  very  large  change  which  is  not  reversed  when 
the  external  conditions  are  restored  to  their  original  values. 
The  system  was  then  in  a  state  of  false  equilibrium.  Dynamical 
illustrations  are  afforded  by  a  weight  maintained  by  friction  on 
a  rough  inclined  plane,  which  may  be  tilted  through  a  small 
angle  without  producing  any  effect,  and  by  a  weight  hanging 
from  a  wire  infinitely  near  its  breaking  point,  when  a  small 
additional  weight,  instead  of  producing  a  small  reversible  elonga- 
tion, snaps  the  wire.  A  physical  illustration  is  afforded  by  a 
superheated  liquid,  which  boils  explosively  on  a  slight  elevation 
of  temperature.  According  to  Gibbs,  systems  in  states  of  false 
equilibrium  are  maintained  by  forces  analogous  to  friction,  called 
l>a$sii-e  resistances},  and  are  to  be  distinguished  from  states  of  true 
equilibrium,  in  which  the  active  forces  are  so  balanced  that  the 
slightest  change  of  force  will  produce  motion  in  either  direction, 
as  in  a  frictionless  machine.  The  distinction  is  to  be  recognised 
by  appeal  to  experience,  and  it  is  only  in  cases  where  the  limits 
of  operation  of  the  passive  resistances  are  closely  approached 
that  there  will  be  any  difficulty  in  recognising  to  which  type  a 
given  equilibrium  state  belongs. 

States  of  equilibrium  may  also  be  classified  into  states  of 
stable,  unstable  and  neutral  equilibrium,  according  as  the  system 
tends  to  return  to  its  initial  state,  or  to  move  further  away  from 
this  state,  or  simply  to  remain  in  the  altered  state,  when  the 
displacing  force  is  removed.  Dynamical  illustrations  are  afforded 
by  a  sphere  resting  at  the  bottom  of  a  bowl,  on  the  top  of  the 
inverted  bowl,  and  on  a  smooth  table  respectively. 

A  consideration  of  the  same  example  also  illustrates  the  result 
established  in  treatises  on  dynamics  that  the  condition  for 
stable,  unstable,  or  neutral  equilibrium  of  a  mechanical  system 
is  that,  for  any  small  displacement  which  does  not  violate  the 
constraints,  the  change  of  potential  energy  shall  vanish  to  the 
first  order,  and  be  positive,  negative,  or  zero  respectively  to  the 
second  order.  When  the  system  is  in  stable,  unstable,  or  neutral 
equilibrium,  the  potential  energy  is  a  minimum,  a  maximum,  or 
stationary  respectively  (Theorem  of  Dirichlet).  Thus  the  work 
done  by  the  system  in  any  infinitesimal  displacement  is  zero  to 
the  first  order,  and  negative,  positive,  or  zero  to  the  second  order, 
for  the  three  cases.  All  these  conditions  refer  only  to  a  par- 


92  THERMODYNAMICS 

ticular  state,  since  the  displacements  are  infinitesimal,  and  it 
does  not  follow  that  any  maximum  or  minimum  value  is  the 
greatest  or  least  value  of  the  potential  energy  respectively  but 
only  greater  or  less  than  all  other  values  in  the  immediate 
neighbourhood  (cf.  H.  M.,  §  22). 

Physical  examples  of  the  three  types  are  afforded  by  a  gas 
contained  in  a  cylinder  under  an  external  pressure  equal  to  the 
gas  pressure,  by  a  superheated  liquid,  and  by  a  mixture  of  water 
and  saturated  steam,  under  the  same  conditions  respectively. 

50.    Conditions   for   Equilibrium,  and   for    Stability   of    Equi- 
librium. 

In  the  investigation  of  the  equilibrium  states  of  therrnodynarnic 
systems  there  are  two  points  of  departure,  which  are  really  more 
or  less  equivalent. 

The  first  is  Lord  Kelvin's  principle  of  Dissipation  of  Energy 
(1852)  which  was  generalised  and  applied  to  chemical  reactions 
by  Lord  Rayleigh  (1875).  According  to  this,  any  change  (and 
in  particular  a  chemical  reaction)  is  impossible  if  it  leads  to  the 
reverse  of  dissipation  of  energy,  or,  as  we  may  call  it,  to 
motivation  of  energy.  In  the  application  of  this  criterion  we  have 
to  determine  how  the  available  energy  of  the  system  depends  on 
the  variables  defining  its  state,  and  we  can  then  find  whether 
any  imaginary  change  in  the  state  of  the  system  which  does  not 
violate  the  given  conditions  {e.g.,  of  constant  volume,  or  constant 
temperature,  or  constant  pressure,  etc.)  leads  to  an  increase,  or 
to  a  decrease,  of  the  available  energy.  If  to  the  former,  the 
imaginary  change  (which  we  shall  call  a  virtual  change)  cannot 
be  realised,  and  we  can  certainly  infer  that  the  system  is  in 
equilibrium ;  but  if  to  the  latter,  we  cannot  say  whether  the 
system  will,  or  will  not,  undergo  the  change  leading  to  dissipa- 
tion of  energy,  because  it  may  be  in  a  state  of  false  equilibrium. 
Thus  we  should  find  that  gunpowder  ought  not  to  exist  at  all 
under  ordinary  conditions,  because  the  explosion  of  gunpowder 
leads  to  dissipation  of  energy ;  the  fact  that  gunpowder  and  such 
materials  do  exist  is  a  consequence  of  the  existence  of  states  of 
false  equilibrium.  The  same  holds  good  with  respect  to  hundreds 
of  organic  compounds  (e.a.,  diazo-compounds). 

The  second  general  principle  is  based  on  the  properties  of  the 
entropy  function,  and  is  contained  in  the  aphorism  of  Clausius 


THERMODYNAMIC  FUNCTIONS  AND  EQUILIBRIUM    93 

(1865)  that  "the  entropy  of  the  universe  tends  to  a  maximum." 
This  is  closely  connected  with  the  principle  of  dissipation  of 
energy,  but,  as  we  saw  in  §  48,  is  not  wholly  co-extensive  with  it. 
By  "  universe "  we  are  to  understand  an  isolated  system,  and 
Clausius's  theorem  is  more  sharply  expressed  in  the  statement 
(§  46)  that,  in  all  real  changes  occurring  in  such  a  system,  the 
entropy  can  only  increase.  If  the  system  has  reached  such  a 
state  that  any  virtual  change,  which  does  not  violate  the  con- 
dition of  constant  energy,  leads  to  a  decrease  of  entropy,  the 
system  is  certainly  in  equilibrium,  for  then  no  change  is  possible. 
This  method  was  first  applied  to  chemical  problems  by  A. 
Horstmann  (1873). 

51.     Gibbs's  Two  Criteria  of  Equilibrium. 

(1)  For  the  equilibrium  of  an  isolated  system  it  is  necessary  and 
sufficient  that  in  all  possible  variations  of  the  state  of  the  system 
which  do  not  alter  its  energy  the  variation  of  its  entropy  shall  either 
vanish  or  be  negative. 

For  in  all  actual  changes  of  such  a  system  the  entropy  can 
only  increase,  so  that  if  we  consider  a  virtual  change,  and  put 
(5S)u  for  the  resulting  change  of  entropy,  then : 

if  (6S)u  >  0  the  change  is  possible  and  irreversible, 
if  (§S)u  <  0  the  change  is  impossible, 

whilst  if  (SS)(j  =  0  the  change  is  reversible. 

The  condition  for  equilibrium  is,  therefore, 

(8S)F<0  .         .      (1) 

for  the  change  represented  by  the  inequality  is  impossible,  whilst 
that  represented  by  the  equality  is  reversible,  and  reversible 
changes  can  only  occur,  as  a  limiting  case,  when  the  system  is 
in  equilibrium. 

The  equilibrium  is  evidently  stable  when  the  entropy  is  a 
maximum,  for  then  every  possible  change  would  diminish  the 
entropy.  The  equilibrium  will  be  unstable  when  the  entropy  is 
a  minimum  for  a  given  value  of  the  energy.  This  implies  that 
if  there  are  several  conceivable  neighbouring  states  with  the 
same  energy,  that  with  the  least  entropy  will  correspond  with  a 
state  of  unstable  equilibrium,  whilst  the  others  with  more  entropy 
will  be  essentially  unstable  states,  except  the  one  with  the 
greatest  amount  of  entropy,  which  will  be  the  state  of  stable 


94  THERMODYNAMICS 

equilibrium.  Thus,  if  water  vapour  is  cooled  without  admission 
of  liquid  or  nuclei  on  which  liquid  can  condense,  there  is  formed 
homogeneous  vapour  which  at  a  given  temperature  exists  under 
a  pressure  greater  than  the  pressure  of  saturated  vapour.  If 
this  is  now  isolated  we  have  a  state  of  unstable  equilibrium. 
The  entropy  in  this  state  is  less  than  that  in  any  of  the  hetero- 
geneous states  produced  by  separation  of  liquid,  and  all  these 
(except  the  one  in  which  liquid  is  in  contact  with  saturated 
vapour,  which  has  the  maximum  entropy  and  is  the  stable  state) 
are  essentially  unstable  states.  The  validity  of  the  condition 
for  unstable  equilibrium  may  be  rendered  apparent  as  follows. 
Suppose  that,  besides  the  state  of  unstable  equilibrium  a,  there 
could  be  some  other  state  ,8,  which  had  less  entropy  than  a,  for 
the  same  energy.  Then  we  could  arrive  at  a  by  starting  with  /3, 
but  the  system  would  not  remain  in  the  state  a,  because  the 
latter  is  not  a  state  of  maximum  entropy,  and  all  the  changes 
which  carry  the  system  further  from  that  state  are  possible. 
Hence  /3  cannot  have  less  entropy  than  a,  for  a  has  been  assumed 
to  be  an  equilibrium  state,  and  so  a  is  a  state  of  minimum 
entropy,  with  respect  to  states  in  its  immediate  neighbourhood. 

Again,  if  all  the  states  in  the  immediate  vicinity  of  the  equi- 
librium state  have  the  same  entropy  as  the  latter,  the  equilibrium 
is  neutral,  since  there  is  no  reversible  direction  in  which  the 
entropy  can  increase,  and  hence  no  tendency  to  pass  from  any 
one  of  these  states  to  any  other. 

The  condition  that  the  equilibrium  shall  be  stable,  unstable,  or 
neutral  is  that  the  entropy  shall  be  a  maximum,  a  minimum  or 
stationary  respectively : 

if  (&2S)u  <  0  the  equilibrium  is  stable, 
if  (S2S)u  >  0  the  equilibrium  is  unstable, 
whilst  if  (82S)u  =  0  the  equilibrium  is  neutral. 

The  values  of  82S  are  to  be  calculated  in  the  usual  way  by 
Taylor's  theorem. 

(2)  For  the  equilibrium  of  any  isolated  system  it  is  necessary  and 
sufficient  that  in  all  possible  variations  of  the  state  of  the  system 
which  do  not  alter  its  entropy,  the  variation  of  its  energy  shall 
either  vanish  or  be  positive : 

(aU)s>0         .".      •  .     .  •:.  ,      .     (2) 

Criterion  (2)  is  the  antithesis  of  criterion  (1),  and  the  validity 
of  the  one  implies  that  of  the  other.  For  it  is  always  possible  to 


THEPvMODYNAMIC   FUNCTIONS  AND  EQUILTBEIUM    95 

increase  or  decrease  the  entropy  and  energy  of  a  system  together 
by  the  addition  or  abstraction  of  heat.  Then  if  criterion  (1)  is 
not  satisfied,  there  will  be  some  variation  for  which : 

5S  >  0  and  8U  =  0 ; 

hence,  by  diminishing  both  the  entropy  and  the  energy  of  the 
system  in  its  altered  state,  we  shall  obtain  a  state  for  which, 
considered  as  a  variation  of  the  initial  state : 

SS  =  0  and  8U  <  0, 
which  does  not  satisfy  criterion  (2). 

Conversely,  if  (2)  is  not  satisfied,  there  must  be  some  variation 
for  which : 

SU  <  0  and  SS  -  0, 
which  does  not  satisfy  criterion  (1). 

Further,  the  equilibrium  will  be  stable,  unstable,  or  neutral, 
according  as  the  energy  is  a  minimum,  a  maximum,  or  stationary 
respectively,  that  is : 

if  (e>2U)s  >  0  the  equilibrium  is  stable, 
if  (82U)S  <C  0  the  equilibrium  is  unstable, 
whilst  if  (S2U)S  =  0  the  equilibrium  is  neutral. 

These  conditions  may  be  deduced  similarly  to  those  in  the 
first  case. 

Criterion  (1)  is  seen  to  be  identical  with  Horstmann's  principle; 
it  has  been  largely  employed  in  the  treatment  of  equilibria  by 
Planck.  It  is,  however,  not  always  convenient  in  application 
because  the  systems  which  actually  occur  in  practice  are  not 
isolated ;  we  shall  therefore  modify  the  relation  so  as  to 
make  it  suitable  for  non-isolated  systems.  In  this  investigation 
we  shall  recover  the  first  general  method  for  determining 
the  conditions  of  equilibrium — the  principle  of  dissipation  of 
energy. 

52.     Isothermal  Conditions:   Free  Energy  and  Potential. 

"  We  have  (§  45)  established  the  relation 

<S2-Si (1) 

for  changes  occurring  in  any  thermodynamic  system,  Si  and  83 
beiog  the  entropies  of  the  system  in  its  initial  and  final  states, 
and  SQ  an  element  of  heat  absorbed  from  a  body  at  temperature 
T.  The  integral  is  taken  over  the  change  from  the  initial  to  the 


96  THERMODYNAMICS 

final  state,  and  the  sign  of  summation  includes  all  the  bodies 
from  which  heat  is  taken.  (1)  can  be  written : 

where  w  is  a  quantity  which  is  always  positive  in  real  changes, 
but  tends  to  the  limit  zero  in  the  limiting  case  of  reversibility. 
Evidently,  <«>  is  the  uncompensated  increase  of  entropy  of  the 
system  (§  47). 

//'  the  change  is  isothermal,  T  =  constant,  and 

'.-.  QT  =  T(Sa-Si)-T«  .  .  .  (8) 
The  magnitude  on  the  left  is  the  heat  absorbed  in  the  isothermal 
change,  and  of  the  two  expressions  on  the  right  the  first  is 
dependent  only  on  the  initial  and  final  states,  and  may  be  called 
the  compensated  heat,  whilst  the  second  depends  on  the  path,  is 
always  negative,  except  in  the  limiting  case  of  reversibility,  and 
may  be  called  the  uncompensated  heat  From  (3)  we  can  derive 
the  necessary  and  sufficient  condition  of  equilibrium  in  a  system 
at  constant  temperature. 

Then,  either  no  change  at  all  can  occur,  or  all  possible  changes 
are  reversible.  Hence,  if  we  imagine  any  isothermal  change  in 
the  state  of  the  system,  and  calculate  the  value  of  Too  for  that 
change,  this  value  will  be  positive  or  zero  if  the  former  state  is 
an  equilibrium  state. 

Now  QT  =  U2  —  Ui  +  AT       .         .         .     (4) 

/.  (Ua  -  Ui)  -  T  (Sa  -  SO  +  AT  =  -  To> .         .     (5) 

Since  the  condition  for  the  equilibrium  state  involves  neither 
U  nor  S  separately,  but  only  the  magnitude  (U— TS),  we  may 
put: 

U  —  TS  =  *        .        .        .        .     (6) 

where  *  is  a  continuous  and  uniform  function  of  the  state  of  the 
system,  and  was  called  by  Helmholtz  the  Free  Energy.  That  * 
has  the  dimensions  of  energy  is  evident  from  (5),  and  the 
appropriateness  of  the  epithet  "  free  "  will  appear  immediately. 
Then  (5)  can  be  written  : 

*a  —  *i  +  AT  =  —  To>         .         .         .     (7) 
since  *x  =  (Ui  -  TSi),  and  *2  =  (U2  —  TS2). 

U  and  S  contain  arbitrary  terms,  say  a  and  /3,  depending  on 
the  choice  of  the  initial  states  of  zero  energy  and  entropy 
respectively ;  hence  *  will  contain  an  arbitrary  linear  function  of 


THERMODYNAMIC  FUNCTIONS  AND  EQUILIBRIUM     97 

temperature,  a  —  /3T,  which  does  not,  however,  enter  (5)  or  (7), 
these  involving  only  differences  of  4>. 

Since  o>  is  essentially  positive  for  all  real  changes,  (7)  shows 
that  in  all  real  isothermal  changes  the  magnitude  %  —  ^i  +  AT 
must  diminish,  and  that  in  any  small  virtual  change  : 

(d  *  +  8A)r  =  —  Trfo>         .         .         .     (8), 
so  that  if  the  system  is  imagined  to  undergo  such  a  change 

and  if  (dV  +  8A)r  <  0  the  change  is  possible  and  irreversible, 
(9)  if  (d*  +  SA)r  >  0  the  change  is  impossible, 

whilst  if  (rf*+  SA)T=:  0  the  change  is  reversible. 
The  criterion  of  equilibrium  of  a  system  maintained  at  constant 
temperature  is  therefore  : 

(d*  +  6A>r  >  0  .         .         .         .     (10) 
for  all  virtual  isothermal  changes. 

If  8AT  >0,  i.e.,  work  is  done  by  the  system,  then  d*  <  0  in  all 
real  changes  ;  if  8AT  <  0,  i.e.,  work  is  spent  upon  the  system, 
then  d*  is  either  >  0  or  <  0  ;  whilst  if  8AT  =  0,  i.e.,  the  system  is 
mechanically  isolated  and  the  virtual  change  is  adynainic  as  well 
as  isothermal,  then  d*  <  0  in  all  real  changes  and  <7*  =  0  in  the 
limiting  case  of  reversible  changes. 

The  necessary  and  sufficient  criterion  of  equilibrium  in  a 
mechanically  isolated  system  at  a  given  temperature  is  : 

WT,,>0       ....     (11) 
for  all  virtual  isothermal  adynamic  changes. 

The  suffix  x  indicates  that  besides  T,  all  the  variables  ?\,  r*,  .  .  .  during 
the  change  of  which  external  work  is  done,  are  maintained  constant 
(adynamic  condition).  Thus,  if  the  only  external  force  is  a  normal  and 
uniform  pressure  p,  then  x  =  v,  the  volume  of  the  system,  and  (11)  is  the 
condition  of  equilibrium  at  constant  temperature  and  volume. 

The  equilibrium  is  stable,  unstable,  or  neutral,  according  as  * 
is  a  minimum,  a  maximum,  or  stationary  respectively  ;  hence  : 
if  (o2*)r,;r  >  0  the  equilibrium  is  stable      } 
if  (o2*)^  <  0  the  equilibrium  is  unstable  r  .     (12) 
whilst  if  (oH)^  =  0  the  equilibrium  is  neutral   ; 


The  equation  for  reversible  isothermal  changes  : 

(d*  +  2oA)r  =  0 

shows  that  —  d*T  =  28AT  ....     (13), 

so  that  in  such  changes  the  external  work  is  done  wholly  at  the 
cost  of  the  free  energy  of  the  system.   The  appropriateness  of  the 


98  THERMODYNAMICS 

latter  name  is  thus  apparent  ;  *  represents  that  part  of  the 
intrinsic  energy  which  is  freely  available  for  conversion  into  exter- 
nal work  in  isothermal  processes  ;  it  is  the  available  energy  at  con- 
stant temperature,  and  corresponds  exactly  with  the  potential 
energy  of  mechanical  systems.  We  may  therefore  speak  of  a 
system  as  possessing  a  charge  of  free  energy,  readily  capable  of 
being  realised  as  useful  external  work  in  isothermal  changes,  just 
as  a  bent  bow  possesses  a  store  of  potential  energy  realisable  as 
the  kinetic  energy  of  a  discharged  arrow.  The  free  energy  is 
assigned  to  a  system  on  the  condition  that  the  changes  in  which 
it  is  realised  are  isothermal  and  reversible.  We  can  of  course 
say  that  in  irreversible  changes  that  part  of  the  free  energy  which 
is  not  rendered  available  has  been  lost  by  dissipation,  e.g.,  con- 
verted into  heat  ;  and  equations  (9)  show  that  in  irreversible 
changes  —  d^>T  >  25AT,  i.e.,  the  external  work  is  less  than  the 
diminution  of  free  energy. 

Equation  (13)  gives  on  integration  : 


Now  (6)  can  also  be  written  in  the  form  : 

U  =  *  +  TS      .        .         .        .     (14), 

so  that  in  the  sense  explained  we  may  regard  the  total  intrinsic 
energy  as  made  up  of  two  parts  : 

(i.)  The  free  energy  SP,  which  is  available  energy  in  reversible 
isothermal  changes  ; 

(ii.)  The  part  TS,  which  is  unavailable  energy  in  reversible 
isothermal  changes,  and  was  called  by  Helmholtz  the  Bound 
Energy  B,  of  the  system. 

The  equation 

(Ua  -  Ui)  =  (*a  -  *0  +  (Ba  -  BI)   ) 
or  (Ua  -  Ui)  =  (*a  -  *i)  +  T(S2  -  Si)' 

shows  that  the  total  loss  of  intrinsic  energy  in  any  isothermal 
reversible  change  in  the  system  is  the  sum  of  the  changes  of 
the  freely  available  part,  or  *x  —  *2,  and  of  the  unavailable  part, 
viz.,  BI  —  B2,  which  latter  is,  as  we  see  from  the  relation,  B  =  TS, 
or 

Ba-Bi  =  T(Sa-Si)  .  .  .  (16), 
given  off  as  heat  to  the  constant  temperature  medium  surround- 
ing the  system.  (*:  -  *2)  and  (Bi  -  B2)  are  called  by  Haber 
(Thermodynamics  of  Technical  Gas  Reactions)  the  reaction  energy 
and  latent  heat  respectively,  when  they  refer  to  chemical  changes. 


THERMODYNAMIC  FUNCTIONS  AND  EQUILIBRIUM    99 

The  equation  (16)  shows  that  the  increase  of  hound  energy  in 
a  reversihle  isothermal  change  is  equal  to  the  increase  of  entropy 
multiplied  hy  the  absolute  temperature,  so  that  the  entropy  may 
be  regarded  as  the  capacity  for  bound  energy  in  such  changes. 
B  will  evidently  contain  the  arbitrary  term  /3T. 

As  soon  as  we  had  shown  that  ¥  is  an  available  energy,  from  the  defini- 
tion of  §  39  and  equation  (13a),  we  could  at  once  have  inferred  the  relations 
(10) — (12)  from  the  principle  of  dissipation  of  energy,  for  ¥  must  be  a  mini- 
mum in  stable  equilibrium. 

In  the  investigation  of  the  properties  of  the  free  energy  ^  no 
assumption  has  been  made  as  to  the  nature  of  the  external  work 
AT.  Let  us  now  assume  that  there  is  some  function  £>  of  the 
variables  defining  the  physical  and  chemical  state  of  the  system, 
such  that : 

£>:-il2  =  AT, 

so  that  the  external  forces  exerted  by  the  system  on  exterior  bodies 
have  &  force-potential  Q.  (cf.  H.  M.,  §  114). 

Thus  (7)  becomes : 

(*a  -  *i)  +  (Si  -  £2)  =  -  To,          .         .     (18), 
or  if  we  put  *  —  £  =  <J>        .         .         .         .     (19), 

we  shall  have : 

4>2  —  *!  =  —  Tco  .         .         .         .     (20), 

so  that  4>  can  only  diminish  in  real  isothermal  changes.  4>  is 
called  by  Duhem  (Traite  de  Mecanique  chimique,  I.,  90)  "  the 
thermodynamic  potential  for  a  given  force-function,"  or  "  the 
total  thermodynamic  potential "  (in  contrast  to  *,  which  he  calls 
"  the  internal  thermodynamie  potential  "). 

A  very  important  case  is  that  in  which  the  sole  external 
force  is  a  uniform  normal  and  constant  pressure  j>. 

Then  AT=-p(r2—  i'i) 

.-.  Q.  =  -  pr      .         .        .         .     (21) 
and  4>P,T  =  *  +  pr  =  U  —  TS  +  pr        .        .     (22 ) 

We  shall  often  denote  ^,.T  simply  by  <£,  so  that : 

<f>  =  U  -  TS  +  pr  =  *  +  pv  .  .  .  (23) 
and  call  <£  the  thermodynamic  potential  for  constant  pressure,  or 
simply  the  potential,  of  the  system. 

Thus  4>2  —  <#>!=  —  Tco    .         .         .         .    (24), 

so  that  in  any  real  isothermal  and  isopiestic  change  </>  can  only 
diminish. 

From  this  and  (20)  we  can  deduce  the  criteria  for  equilibrium, 

H  2 


100  THERMODYNAMICS 

and  for  stability  of  equilibrium,  of  a  system  maintained  at  con- 
stant temperature  and  constant  external  forces  (X),  in  particular 
at  constant  pressure  (p).  For,  if  the  system  undergoes  a  virtual 
isothermal  change  with  constant  external  forces,  and  (8<J>)T,x  is 
the  change  of  *, 
/'then  if  (8*)r,x  <  0  the  change  is  possible  and  irreversible, 

if  (8*)r,x  >  0  the  change  is  impossible, 
(whilst  if  (84>)T,x  =  0  the  change  is  reversible. 

Hence,  for  equilibrium  in  a  system  maintained  at  constant 
temperature  and  with  constant  external  forces,  it  is  necessary  and 
sufficient  that  for  all  virtual  changes  : 

(S4>)T,x>0       ....     (25) 
The  equilibrium  is : 

stable  if  (S2d>)T)x  >  0, 

unstable  if  (824>)T')X  <  0,        .         .         .     (26) 
neutral  if  (S2<I>)Tlx  =  0. 

We  observe  that  these  criteria  could  at  once  have  been  deduced  from  the 
principle  of  dissipation  of  energy  after  we  had  established  that  *  is  an 
available  energy,  from  (13),  (17),  and  (19).  Similarly  for  the  conditions 
referred  to  $,  which  follow. 

If  the  only  external  force  is  a  normal,  uniform,  and  constant 
pressure  p,  the  necessary  and  sufficient  condition  for  equilibrium 
is  that  for  all  virtual  isothermal-isopiestic  changes  : 

WT)3)>0       ....     (27) 

whilst  the  equilibrium  is  stable,  unstable,  or  neutral,  according 
as  : 

(S2</>)/()T  >  0,  (S2<a,T  <  0,  or  (S2<a,T  =  0         .     (28) 
respectively. 

If  there  are  several  maxima  of  entropy,  or  minima  of  available  energy,  there 
will  be  several  equilibrium  states  for  the  system,  but  the  one  corresponding 
with  the  greatest  of  the  maxima,  or  the  least  of  the  minima,  will  be  the  most 
stable  state.  Further,  if  for  a  given  state  there  is  a  series  of  changes 
for  which  80  >  0  another  for  which  5$.  =  0,  and  yet  another  for  which 
8<£  <  0,  then  if  the  system  is  given  an  "  initial  impulse  "  it  will  pass  along 
the  series  of  changes  for  which  5<?>  <  0,  so  that  the  determined  equilibrium 
state  is  not  absolute,  but  only  relative  to  particular  changes.  Thus,  if  a 
system  is  in  stable  equilibrium  with  respect  to  isothermal  changes,  it  can  be 
shown  to  be  in  stable  equilibrium  with  respect  to  isentropic  changes,  but  the 
converse  is  not  necessarily  true. 


THERMODYNAMIC  FUNCTIONS  AND  EQUILIBRIUM     101 

53.     Characteristic  Functions. 

The  functions  : 

¥  =  U-TS    .......         (1) 

*  =  U  -  TS  +  pv  =  V  +  2,v          .         .        .        (2) 

have  an  application  in  thermodynamics  which  far  transcends  their  utility  in 
the  study  of  isothermal  changes.  They  have  been  used  by  various  authors 
under  different  names,  and  it  seems  desirable  to  mention  these,  so  that  the 
reader  will  have  no  difficulty  in  recognising  the  functions  in  the  literature  of 
the  subject. 

F.  Massieu  (Journ.    de   Plnjs.    4,    216,    1869)  first    showed  that  all  the 
characteristic  properties  of  fluids  can  be  expressed  in  terms  of  one  or  other 
of  two  functions,  H  and  H'  defined  as  follows  : 
_  -  U  +  TS  _       y      ,>_ 


_  _ 

T  T  '  '  T  T' 

and  he  called  these,  referred  to  unit  mass,  the  characteristic  functions  of  the 
fluid. 

J.  Clerk  Maxwell  ("  Theory  of  Heat,"  1871)  observed  that  the  maximum 
work  of  an  isothermal  process  could  be  represented  as  a  diminution  of  a 
function  U  —  TS,  which  he  at  first  called  the  "  entropy,"  but  afterwards 
(1875)  the  available  energy.  This  latter  name  had  been  proposed  by  J.  "\Vil- 
lard  Gibbs  (1873)  in  a  very  important  memoir  on  "A  Method  of  Geometrical 
Eepresentation  of  the  Thermodynamic  Properties  of  Substances  by  Means 
of  Surfaces"  (Scientific  Papers,  I,,  33).  In  this  memoir  Gibbs  shows  that  the 
conditions  of  equilibrium  of  two  parts  of  a  substance  in  contact  can  be 
expressed  geometrically  in  terms  of  the  position  of  the  tangent  planes  to  the 
volume-entropy-energy  surface  of  the  substance,  and  he  finds  that  the  analyti- 
cal expression  of  this  property  is  that  the  value  of  the  function  (U  —  TS  -f-  pv) 
shall  be  the  same  for  the  two  states  at  the  same  temperature  and  pressure. 
In  his  later  memoir  "On  the  Equilibrium  of  Heterogeneous  Substances  " 
(Trans.  Connecticut  Acad.,  HL,  108  —  248;  343—524  ;  Silliman'sJoitrn.,  16,441, 
1878)  the  three  functions 

i^  =  U  —  Ts  ("  force  function  for  constant  temperature  "), 
£  =  IT  —  Is  -\-pv  ("  force  function  for  constant  pressure  "), 
X  =  U  +  pv  ("  heat  function  for  constant  pressure  ") 

are  constantly  used,  and  they  are  frequently  referred  to  as  the  psi,  zeta,  and 
chi  functions  of  Gibbs.  The  zeta  function  is  identical  with  our  </>,  or  the 
potential,  the  psi  function  with  the  free  energy,  whilst  the  x  function  is 
the  heat  function  at  constant  pressure  of  §  25. 

In  1879  Lord  Kelvin  introduced  the  term  motivity  for  "  the  possession, 
the  waste  of  which  is  called  dissipation  "  ;  at  constant  temperature  this  is 
identical  with  Maxwell's  available  energy.  He  showed  in  a  paper  "  On 
Thermodynamics  founded  on  Motivity  and  Energy"  (Phil.  Mag.,  1898),  that 
all  the  thermodynamic  equations  could  be  derived  from  the  properties  of 
motivity  which  follow  directly  from  Carnot's  theorem,  without  any  explicit 
introduction  of  the  entropy. 

Helmholtz  (18-82)  generalised  the  potential  energy  function  of  mechanics 
so  as  to  obtain  a  function  which,  at  a  given  temperature,  should  represent 


102  THERMODYNAMICS 

the  maximum  obtainable  work  for  a  given  change  of  configuration,  and 
could  be  transferred  to  other  temperatures  by  means  of  the  second  law  of 
thermodynamics.  This  function  was  the  free  energy,  V. 

Planck  (Thermodynamics,  trans.  Ogg)  has  used  the  second  of  Massieu's 
functions : 

TT'  V  +  pv  * 

-T~    ~r 

which  is  now  usually  called  "  Planck's  Potential."  In  some  cases  this  makes 
the  equations  more  symmetrical  than  those  with  <t>,  and  in  addition  has  the 
same  properties  at  constant  temperature  and  pressure  in  a  non-isolated 
system  as  the  entropy  function  in  an  isolated  system. 

It  must  be  remembered  that  all  these  functions  were  introduced  for  the 
purpose  of  simplifying  the  mathematical  operations,  just  as  were  the  energy 
and  entropy  functions  in  the  earlier  stages  of  thermodynamics.  It  is  only 
their  changes  which  admit  of  physical  measurement ;  these  changes  can  be 
represented  as  quantities  of  heat  and  external  work. 

In  what  follows  we  shall  denote  the  energy,  volume,  and 
entropy  per  unit  mass  by  the  small  letters  u,  r,  s. 

54.     The   Fundamental   Differential   Equations. 

We  shall  in  the  first  place  assume  that : 

(i.)  The  only  external  force  is  a  normal  and  uniform  pressure p  ; 
(ii.)  The  state  of  unit  mass  is  completely  denned  in  terms  of 
two  independent  variables  x  and  y.     (This  does  not  require  that 
the  system  be  homogeneous.) 
We  have  then  the  equations : 

SQ  =  du  +  SA (1) 

8A=pdv (2) 

SQ  =  Tds  (reversible  change)       .         .         .     (3) 

/.   du  =  Tds  —pdv (4) 

Since  du  and  ds  are  perfect  differentials  : 

*•**&*•  +  £*»      ....     (5) 
^-|^  +  |^       ....     (6) 
hence        ^  =  (T*  -p  |)  dx  + 

But  du  is  also  a  perfect  differential,  from  the  first  law  : 


THEKMODYXAMIC  EQUATIONS  AND  EQUILIBRIUM     103 


y  cy        f  dy 

By  Euler's  criterion  (H.  M.,  §  57)  : 


oy 

cT  cs^  _  cT  QJ  _  c/^cr  _  fy>  cr 
'  ex   cy       oy    dx        d.c  cj/        c#  cr 

This  is  the  fundamental  differential  equation.  The  reader  who  is  ac- 
quainted with  the  rules  for  transforming  the  variables  in  a  surface  integral 
will  observe  that  it  has  the  geometrical  interpretation  that  corresponding 
elements  of  area  on  the  (r,  p]  and  (s,  T)  diagrams  are  equal  (cf.  §  43). 

Of  the  five  magnitudes  p,  v,  t,  s,  u,  any  two  may  be  chosen 
for  the  independent  variables  x  and  y,  and  for  each  pair  we 
shall,  by  means  of  (11),  obtain  a  relation  between  the  differential 
coefficients  of  two  other  magnitudes  with  respect  to  the  chosen 
variables.  These  are  deduced  as  follows  : 

(1)  Let  x=s,y=v: 

cT     cs       ?T  _  cp       op     cr 
cs     cv       cr       cs        cr  '  cs 

But  —  =  0  and  —  =  0,  because  r  and  s  are  by  hypothesis 
cr  cs 

independent  variables, 

.-.-(£)  =  («)      .     .     .  CD 

\csJc       \cr/s 

(2)  Let  x  =  v,  y  =  T  : 

3T     cs        cs  _  cp     cr         cp 

*"•  aF  '  er  ~  ?F  "~  ?7  '  ?T  ~  er* 

But  ^  =  0,  and  ^'  =  0, 

••• 

(3)  Let  x  =  s,  y=p: 

1^    fl  _  ^  —  §P     fi"  _  £i" 
*  *-  cs      c/>         cji         cs  '  cp        cs 

But  |^  =  0,  and  -?  =  0 

CJJ  CS 

.-.  (f.T)  =  (?)     ...  (III.) 

\Cpls  \CS/p 

(4)  Letj;  =  T,?/=7j: 

cs        8T     3s  _  cjj     cr        cr 
"    3p       8p  *  9T       3T  *  8p       3T 


104  THERMODYNAMICS 


But       =  0.  and       =  0, 

'    -(*-)    -(-)  (TV) 

\dpJT~  V3T/, 

The  four  relations  (I.)  —  (IV.)  are  usually  known  as  MaxweWt 
Relations,  or  the  Reciprocal  Relations  ;  they  were  deduced  by 
Maxwell  by  means  of  an  ingenious  geometrical  method  (Theory 
of  Heat,  Chap.  9). 

Exercises  on  the  Reciprocial  Relations. 

(1)  The  fall  of  temperature  per  unit  increase  of  volume  in  adiabatic  expan- 
sion is  equal  to  the  increase  of  pressure  per  mechanical  unit  of  heat  supplied 
at  constant  volume,  multiplied  by  the  absolute  temperature. 

[Multiply  and  divide  the  left  hand  member  of  (I.)byT,  and  put  'Ids  = 
80.] 

(2)  The  increase  of  pressure  per  one  degree  rise  of  temperature  at  constant 
volume,  multiplied  by  the  absolute  temperature,  is  equal  to  the  heat  absorbed 
per  unit  increase  of  volume  at  constant  temperature. 

(3)  The  rate  of  increase  of  volume  per  mechanical  unit  of  heat  absorbed  at 
constant  pressure  is  equal  to  the  adiabatic  rate  of  rise  of  temperature  with 
pressure,  divided  by  the  absolute  temperature. 

(4)  The  heat  evolved  per  unit  isothermal  increase  of  pressure  is  equal  to 
the  continued  product  of  the  absolute  temperature,  the  specific  volume,  and 
the  coefficient  of  expansion  o. 

(-HIX-) 

(5)  Show  that  : 


(6)  Show  that  : 

(dp\  (dv\          /a  A  pA 

W*var/,     VsT/Aas 

(7)  Prove  the  relations  : 

/ar\  /aA       /ar 


(8)  The  diminution  of  energy  per  unit  increase  of  volume  at  constant 
entropy  (i.e.,  in  adiabatic  changes)  is  measured  by  the  pressure. 

(9)  The  increase   of   energy  per  unit  increase   of   entropy  at  constant 
volume  is  measured  by  the  absolute  temperature. 

Corollary.  —  At  constant  volume,  the  gain  of  energy  is  measured  by  the 
heat  absorbed. 


THERMODYNAMIC  EQUATIONS  AND  EQUILIBRIUM  105 
55.  Free  Energy  and  Potential. 

Still  considering  the  system  of  the  preceding  section,  we  intro- 
duce the  free  energy  ^  and  potential  <j>  of  unit  mass  into  the  list 
of  variables  : 

p,  r,  T,  s,  «,  Vr,  <£. 
If  we  put  T  =  const,  in  the  fundamental  equation  : 

rfU  =  SQ  -  SA 
we  have  for  isothermal  changes  : 

dtt  =  d(Ts)  —  SA 

so  that  8  A  is  in  this  case  a  perfect  differential,  —  d(u  —  Ts)  (cf.  §  36). 
Now  8  A  =  pdr,  whence  it  follows  that  in  isothermal  changes,^ 
can  be  expressed  as  a  function  of  v  alone,  and  F(c)dc  is  a  perfect 
differential. 

From  the  equation 

dn  =  Tds  —  pdr 
we  have,  on  taking  r  and  T  as  independent  variables  : 


•  u> 


In  isothermal  changes  T  =  const. 

.  T  (^\    - 

"  L  ~ 


•, 


V&r/  T        V    dv  /  T       9f  \?r  /  T 

which   give  the  entropy  and  pressure  in   terms  of  differential 
coefficients  of  the  free  energy. 

Now  take  p  and  T  as  independent  variables  : 


106  THERMODYNAMICS 


*)  +  „(<>)  _T(?)    =o 

9p/  T  \C^/  T  VpJ  T 


8p/  T 

/7r\  /d(?H-)\ 
and                                _p  (  ^-       =     -5*-J  )     — 

1      \8p/T  V    op    /T 

._  /3(T«) 


..  _         _  P  =  p  =  -  «     •         •     (5) 

^O,_TS+^)TE(^)   =r  .  (6), 

op  v  \9p/  T 

which    give  the    entropy  and  specific    volume    as    differential 
coefficients  of  the  potential. 

Since  the  entropy  is  the  partial  differential  coefficient  of  ^ 
with  r  constant,  or  $  with  p  constant,  with  respect  to  T,  the 
magnitudes  ^  and  <f>  are  often  called  the  thermodynamic 
potentials  at  constant  volume  and  at  constant  pressure  respec- 
tively. 

By  combining  the  equations  : 

u  =  +  +  T« 


we  find  the  very  important  equation  : 


which  is  called  the  Free  Energy  Equation  (Helmholtz,  1882). 
Similarly,  from  the  equations 

u  =  <£  +  Ts  —  pv 


we  obtain  the  equally  important  equations 


THERMODYNAMIC  EQUATIONS  AND  EQUILIBRIUM    107 

Equations  (7)  and  (8)  may  be  multiplied  through  by  the  mass 
m  to  give  the  relations 


U  =  *-T  -PV    •        •        •(&>), 

where  the  large  symbols  refer  to  the  total  energy,  free  energy, 
potential,  and  volume  of  the  given  mass. 

If  SQ  is  the  small  amount  of  heat  absorbed  in  a  small  rever- 
sible igothermal-isopiestic  change,  we  have,  if  W  is  the  heat  function 
at  constant  pressure  : 


SQ  =  rf(U  +  p\)  =  dW  =  d   *  -  T  =  .  (10), 

an  equation  which  is  often  useful. 

56.     Generalisation  ;    Relations    for    Systems     with    Several 
Degrees   of   Freedom. 

We  shall  now  consider  the  properties  of  systems  the  state  of 
which  is  determined  by  the  values  of  the  absolute  tempera- 
ture T,  and  n  other  independent  variables  Xi,  x.2,  x3,  .  .  .  xn. 
If  the  latter  are  chosen  in  such  a  way  that  no  external  work 
is  done  when  the  temperature  changes  provided  all  the  a-'s 
are  maintained  constant,  they,  along  with  T,  are  called  the  normal 
variables,  and  the  state  so  defined  is  said  to  be  normally  defined 
(Duhem:  Mecaniquechimique,  I.,  33). 

The  or's  may  be  said  to  define  the  configuration  of  the  system, 
and  the  normal  variables  therefore  define  the  state  of  the 
system  in  terms  of  its  configuration  and  temperature.  Changes 
of  state  may  then  be  changes  of  configuration  at  constant 
temperature,  or  changes  of  temperature  at  constant  configura- 
tion, or  changes  of  configuration  and  temperature  together. 

Taking  the  simple  case  of  a  homogeneous  fluid  of  unchanging  composition 
we  see  that  its  state  may  be  defined  in  terms  of  any  pair  of  the  three 
variables  :  temperature  T,  specific  volume  i;  and  pressure  p.  If  the  state 
is  to  be  normally  defined,  T  must  be  taken  as  one  variable,  and  v  must  be 
taken  as  the  other,  because  there  is  the  condition  to  be  satisfied  that  no 
external  work  is  done  when  the  temperature  changes  whilst  the  variable 
remains  constant.  This  condition  is  satisfied  by  v,  but  not  by  p. 

The  normal  variables  may,  according  to  the  nature  of  the 
system  considered,  be  the  temperature  and  either  geometrical 


108  THERMODYNAMICS 

magnitudes  such  as  lengths,  areas,  or  volumes,  or  else  physical 
qualities  such  as  the  relative  proportion  of  two  interconver- 
tible phases,  or  a  quantity  of  electricity,  or  the  concentration  of 
a  solution. 

We  now  consider  the  thermodynamic  relations  for  a  system 
which  is  normally  denned.  A  system  having  n  independent 
variables  is  said  to  have  n  decrees  of  freedom. 

For  a  small  change  of  all  the  independent  variables  we  have  : 
SA  =  Xx  dx1  +  X2Jx-2  +  .  .  .  =  SXf<fcr{  .         .     (1), 
since  there  is  no  term  with  dT  when  the  variables  are  normal. 
But         SQ  =  dit  +  SA 


d*i      .         .     (2) 
and          SQ  =  Ids  =  T  (^)  _  dT  +  TS  ( £) _  d,,  .  (3) 


We  can  now  proceed  exactly  as  we  did  when  there  were  only 
two  independent  variables,  v  and  T,  and  find  : 

gfjKtt-    T*)z    =     -*.  .  .  .         (4) 

—K-T^W-.Xi      .       .       .   (5), 
and  if  ^lijra>  .  .  T  =  /^.,2i  .  .  T  -  T^,ia,  .  .T       .         .     (6) 


and  »=-T  ^    ....     (9) 

The  magnitude   (^J^,  or  for  any  mass  (^)  ,  is  the  heat 

capacity  of  the  system  at  constant  configuration,  which  we  shall 
denote  by  Tx,  or  jx  for  unit  mass  : 


which  determines  r^  without  ambiguity,  the  two  arbitrary  con- 
stants in  *  having  been  eliminated  by  differentiation. 
The  bound  energy  is  determined  by  the  equation  : 

B  =  U  -  *  =  -  T  (^)    =  TS      .         .     (11) 


THERMODYNAMIC  EQUATIONS  AND  EQUILIBRIUM     109 

If  the  temperature  is  changed  as  well  as  the  normal  variables, 
the  free  and  bound  energies  alter  as  follows  : 

"*>  =  -  S''T  -  *A  '    <ia> 


=  SrfT  +  SQ   .         .        .     .    .         .        .         .     (13) 

Thus,  during  the  change,  the-  free  energy  diminishes  by  the 
amount  of  the  external  work  &A  and  —  SrfT,  whilst  the  bound 
energy  increases  by  the  amount  of  the  heat  absorbed  8Q  and 
+  S</T.  We  can  therefore  regard  the  amount  SfZT  as  going  over 
from  the  free  to  the  bound  part,  whilst  the  two  latter  diminish  by 
the  work  and  heat  evolved  respectively.  The  only  difference 
between  this  and  an  isothermal  change  is  therefore  the 
transformation  of  the  part  SdT  of  free  energy  into  bound 
energy. 

If  the  free  energy  is  known  at  one  temperature  and  configu- 
ration, it  can  be  calculated  for  all  other  temperatures  and  the 
same  configuration,  provided  we  know  the  heat  capacity  Tx  for 
all  temperatures  in  the  given  range.  For  : 


8T2/  *  \9T 


'  To 

But 


Again, 


-fW  1T1-       f^ 

=  LwJft    ~JToT 


frfT         .        .        .     (14) 
TO 


110  THERMODYNAMICS 

r 

.'.  T0So  -  TiSi  -  (*i  -  *0)  =  -     T,  rfT    .         .     (16) 

J   To 

Multiply  (14)  by  Tj.  and  subtract  (15) 


rTI 

J^-T 


!-*„  =  (To  -TO  So  +   rxi-ciT    .   (ic) 

To 

Coi'ollary  1.  —  If  Tx  is  sensibly  constant  over  a  small  range 
TI—  TO,  we  have  : 

^1_^0=(r,-So)(T1-T0)-r,T1/»^        .     (17) 

We  shall  assume  that  Tx  is  always  positive,  i.e.,  if  the 
temperature  of  a  system  is  raised  at  constant  configuration, 
heat  is  absorbed.  As  Duhem  (Mecanique  cltimiquc,  I.,  164) 
points  out,  this  is  by  no  means  self-evident,  because  there  are 
some  heat  capacities  which  are  negative  ;  the  heat  capacity  at 
constant  configuration,  Tx,  appears,  however,  always  to  be 
positive. 

As  an  example  of  a  negative  heat  capacity  we  have  the  specific  heat  of 
saturated  steam.  If  unit  mass  of  steam  in  the  condition  of  saturation  is 
raised  one  degree  in  temperature,  and  at  the  same  time  compressed  so  as  to 
keep  it  just  saturated  at  each  temperature,  it  is  found  that  heat  is  evolved, 
not  absorbed,  because  the  work  spent  in  the  compression  exceeds  the 
increase  of  intrinsic  energy. 

Finally  we  shall  show  that  the  potential  equation  is  generally 
applicable  to  systems  defined  by  the  independent  variables  : 
T,  Xi,  X2,  X3,        .... 
where  X  denotes  a  generalised  external  action,  or  intensity  : 

SA  =  Xirfxi  +  Xarf-ra  + 

For  <f>T,  X]  X2,  .  -  =  UT.  Xl.  x2.  •  •  -  TST,  Xl,  x2,  +  SX^  .  (1) 
and 


We  observe  that  the  expression  for  the  external  work  now 
involves  the  temperature,  since  the  variables  are  no  longer 
normal. 

If  we  substitute  (2),  (3)  and  (4)  in  the  general  equation  : 
dU  =  SQ  -  SA 


THERMODYNAMIC  EQUATIONS  AND  EQUILIBRIUM     111 

and   proceed   as   in   the   deduction   for   the  simple   case  when 
Xi  =  p,  X2  =  X3  =  .  .  .  =  0,  we  find  : 


ex— 


.         .         .         .     (5) 
.         .         .         .     (6) 
and  U  =  *-T(^(    -SXirfri         ...     (7) 


which  are  identical  in  form  with  the  equations  deduced  for  the 
simpler  case. 

57.     Intensity  and  Capacity  Factors  :   Energetics. 

Clerk  Maxwell  (South  Kensington  Conferences,  1876),  in 
discussing  the  work  of  "Willard  Gibbs,  remarked  that  :  "the 
existence  of  a  system  depends  on  the  magnitudes  of  the  system, 
which  are  :  the  quantities  of  the  components,  the  volumes,  the 
entropies,  as  well  as  on  the  intensities  of  the  system,  viz.,  the 
temperature  and  the  potentials  of  the  components  "  (cf.  §  143). 
In  his  Theory  of  Heat  he  also  refers  to  a  separation  of  the 
variables  in  terms  of  which  the  state  can  be  denned  into  two 
classes,  one  of  which  includes  what  are  called  intensities 
(pressure,  temperature),  and  the  other  magnitudes  (volume, 
entropy). 

A  school  of  physicists  has  arisen  the  chief  doctrine  of  which  is  contained 
in  the  assertion  that  the  measure  of  every  form  of  energy  can  be  expressed 
as  the  product  of  two  factors,  one  of  which  is  an  intensity  (pressure,  mecha- 
nical force,  surface  tension,  electromotive  force,  chemical  intensity,  tempera- 
ture) and  the  other  a  capacity  (volume,  distance,  surface,  quantity  of 
electricity,  mass,  entropy).  The  capacities  determine  what  may  be  called 
the  material  properties  of  the  system  —  its  extension  in  space,  etc.,  and  are 
additive,  i.e.,  if  two  systems  the  capacities  of  which  are  Xi  and  x-2  are  com- 
bined into  a  larger  system,  the  capacity  of  the  latter  is  Xj  -j-  a--2.  Thus,  if 
two  masses  »HI»  n»a  are  placed  together,  the  resulting  mass  is  mi  -f-  »J2  ;  two 
volumes  r1}  v*  produce  a  volume  i'i  +  rs.  The  intensities,  however,  deter- 
mine the  equilibrium  of  the  various  forms  of  energy  in  the  system  ;  the 
hitter  tend  to  pass  from  places  of  higher  to  places  of  lower  intensity,  and  are 
in  equilibrium  when  the  intensity  has  the  same  value  throughout.  They 
are  not  additive  ;  thus,  if  two  vessels  containing  a  gas  under  the  same 
pressure  are  put  in  communication,  the  capacities  (viz.,  the  volumes)  are 
added,  whilst  the  intensity  (viz.,  pressure)  remains  unchanged. 


112  THERMODYNAMICS 

The  further  step  was  then  taken  in  the  assumption  that  in  reality  nothing 
exists  except  energy,  and  all  phenomena  are  energy  transformations. 
Instead  of  considering  the  two  worlds  of  matter  and  energy,  each  with  its 
law  of  conservation,  the  disciples  of  the  school  of  "  Energetics  "—as  this 
doctrine  is  called— prefer  to  regard  matter  as  simply  a  collection  of  energies. 
The  so-called  properties  of  matter  ai%e  certainly  really  those  of  energy, 
since  we  have  no  cognisance  of  the  existence  of  material  objects  except 
through  the  medium  of  the  senses,  and  this  transmission  is  really  a  trans- 
mission of  energy. 

I  have  not  adopted  this  point  of  view  because,  although  apparently  very 
simple  and  plausible,  it  is  not  capable  of  taking  us  much  further  in  the 
physical  interpretation  of  phenomena.  The  Second  Law  of  Thermodynamics, 
so  far  from  being  a  particular  case  of  a  general  "intensity  law," 
according  to  which  the  availability  of  a  charge  of  energy  is  proportional  to 
the  difference  of  the  available  intensity -factors,  is  a  law  which  places  in  a 
clear  light  the  essential  difference  between  heat  energy  and  the  other  forms 
of  energy,  a  difference  which  certainly  has  a  deep  physical  significance,  some 
indications  of  which  have  appeared  in  the  interpretation  given  to  the  law  by 
Boltzmann,  and  applied  with  such  signal  success  to  the  theory  of  radiant 
heat  by  Planck.  That  this  interpretation,  which  I  take  it  is  wholly  foreign 
to  the  system  of  energetics,  is  not  superfluous,  is  abundantly  confirmed  by  the 
many  new  fields  of  research  which  it  has  opened  out,  and  the  character  of 
the  harvest  which  has  been  accumulated  in  the  short  time  between  the 
enunciation  of  the  new  theory  of  "  energiequanta  "  (cf.  Chap.  XVIII.)  and 
the  present  day. 


58.     Kirchhoff's    Equation   and    the   Equation   of    Maximum 
Work. 

Our  problem  is  to  determine  how  the  changes  of  total  and 
free  energy,  AU  and  A*,  or,  what  are  the  same,  the  heat  absorp- 
tion at  constant  configuration  and  the  maximum  work,  Q^.  and 
AT,  of  an  isothermal  and  reversible  process,  alter  with  the 
temperature  of  execution  of  the  process. 

For  this  purpose  we  suppose  the  following  reversible  cyclic 
process  executed.  This  process,  it  will  be  seen,  is  not  a  Carnot's 
cyclic  process,  but  is  of  another  type. 

(1)  Let  the  system  pass  isothermally  and  reversibly  from  the 
initial  state  : 
(a) 
to  the  final  state  : 


This  is  a  change  of  configuration    at   constant   temperature, 
and  the  line  traced  out  on  an  indicator  diagram  if  there  were 


THERMODYNAMIC  EQUATIONS  AND  EQUILIBRIUM     113 

only   one  x  variable   as   abscissa  (e.g.,  x  =  r),   would    be  an 
isotherm. 

(2)  Let  the  temperature  now  be  raised  from  T  to  T  -f-  ST 
whilst   all   the  normal   variables   remain  unchanged   with  the 
values    (fc).      This   is   a    change   of   temperature  at     constant 
configuration,  and  the  line  traced  out  would  be  an  adynamic. 

(3)  Let    the    change    of    configuration    be  annulled   at   the 
infinitesimally   higher   temperature   T  +  oT  by  an   isothermal 
reversible  process  so  that  all  the  normal  configuration  variables 
recover  the  initial   values  (a).      This   is   a   second   isothermal 
process. 

(4)  Let  the  temperature  be  lowered  to  T  whilst  the  configura- 
tion variables  remain  unchanged.     The  system  is,  by  this  second 
adynamic  process,  brought  back  to  the  initial  state. 

The  cycle  process  is  reversible,  since  the  system  can,  with 
suitable  adjustment  of  the  external  forces,  be  kept  always 
infinitesimally  near  its  equilibrium  state  at  every  instant. 

Such  a  cyclic  process,  consisting  of  two  isotherms  and  two 
adynamics,  was  thoroughly  studied  by  Eankine,  and  may  there- 
fore be  called  a  Rankine's  cycle. 

Let  AU  denote  the  change  of  intrinsic  energy,  and  let  Q  be 
the  amount  of  heat  absorbed  at  the  temperature  T,  in  any  part 
of  the  process.  Then,  according  to  the  first  law  : 

2AU  =  2Q-2A  =  0  .  .  .  (1), 
whether  the  process  is  reversible  or  not  ;  and,  according  to  the 
second  law  : 

2§  =  0   .....     (2), 

but  only  when  the  process  is  reversible. 

Let  AU,  AU  +  <7AU  be  the  amounts  by  which  the  intrinsic 
energy  increases  during  the  changes  of  configuration  from  the 
initial  to  the  final  state  at  the  temperatures  T  and  T  +  &T 
respectively,  and  let  r,,  T,  be  the  heat  capacities  at  constant 
configuration  of  the  initial  and  final  states  at  the  mean  tempera- 
ture T  +  |8T.  The  sum  of  the  changes  of  intrinsic  energy  in 
the  four  processes  is  then  : 

AU  +  I>/T  -  (AU  +  </AU)  -  T^T  =  0,  from  (1), 


••    T 

The  change  of  intrinsic  energy  in  an  isothermal  process  increases 

T.  I 


114  THERMODYNAMICS 

per  one  deyree  rise  of  temperature  at  which  the  process  is  executed 
by  an  amount  equal  to  the  difference  between  the  heat  capacities  of 
the  final  and  initial  systems,  both  at  constant  configuration. 

Equation  (3)  was  first  deduced  by  G.  Kirchhoff  in  1854 
(Ostwald's  Klassih-cr,  No.  101). 

Let  the  amounts  of  heat  absorbed  in  the  changes  of  configura- 
tion at  T  and  T  +  ST  respectively  be  Q  and  —  (Q  +  SQ). 

Since  these  processes  are  reversible  and  isothermal,  Q  and 
(Q  -4-  SQ)  depend  only  on  the  initial  and  final  states,  and  are 
the  same  for  different  isothermal  reversible  paths  (Moutier's 
theorem,  §  36). 

The  sum  of  the  amounts  of  heat  absorbed  in  the  four  processes, 
each  divided  by  the  temperature  at  which  it  is  absorbed,  is  : 


_  _     __ 

T      T  +  ST  -~r  T  +  ^ST      T  +  £ST  ~ 
But  r_r 


.  Q     Q  +  SQ  .     gT       _ 

'  T      T  -f  ST  ~r  T  +  £ST  ~ 


or,  proceeding  to  the  limit,  BT  —  »  0, 


?(AU)_T_a/Q 


_ 

8T  3T  \T 

But  AU  =  Q-AT  (6) 

(7\ 

(7) 


where  AT  is  the  maximum  work. 

The  suffix  Ax  shows  that  the  difference  between  the  initial  and 
final  sets  of  normal  variables  is  to  be  maintained  constant.  Thus, 
if  the  only  normal  configuration  variable  is  the  volume,  x  =  v 
and  A.r  =  r2  —  r:  =  Ar,  i.e.,  the  expansion  is  to  be  kept  the  same 
whilst  the  temperature  changes.  Ax-  may  be  said  to  define  the 


of  execution,  or  the  amplitude  of  the  process.     (^}      then 

\  91  /  ±* 

means,  not  the  amount  of  work  done  when  the  temperature  is 
raised  1°,  for  rise  of  temperature  is  effected  adynamically,  rather 
it  denotes  the  amount  of  work  which  must  be  spent  on  the 
system  in  order  to  reverse  at  a  temperature  (T  +  1)  a  process 


THEEMODYNAMIC  EQUATIONS  AND  EQUILIBRIUM     115 


carried  out  isothermally  at  T  between  the  same  limits  of  con- 
figuration, i.e.,  of  the  same  amplitude.  The  initial  tempera- 
ture may  then  be  regained  by  cooling  adynamically  through  1°. 

Equation  (8)  shows  how  the  maximum  work  of  an  isothermal 
process  depends  on  the  temperature  of  execution  of  the  process, 
and  it  may  be  called  the  Equation 
of  Maximum  Work* 

Illustratirc  Example. — Consider 
the   isothermal   expansion   of   an 
ideal  gas  between  fixed  limits  of 
volume : 
x\  =  r,  j'-2  ^—  J-3  •—  •    •.  •  •  — —  0 

If  we  draw  the  two  isochores 
rit'i,  t'zi'zt  on  the  indicator  diagram 
(Fig.  14),  all  the  paths  of  change 
must  lie  within  these  limits,  which 
fix  the  range  or  amplitude  of  the  "' 

process.     Let    the    initial    state 

(ri,  T)  be  a,  and  let  the  gas  expand  isothermally  and  rever- 
sibly  to  a' ;  the  maximum  work  AT  is  represented  by  the  area 
aa'vzi'i,  which  is  shown  later  to  be  : 


if  we  are  considering  a  mol  of  gas.  Now  suppose  we  had 
taken  the  initial  state  at  b,  the  temperature  being  ST  higher. 
For  reversible  isothermal  expansion  of  the  same  amplitude,  viz., 
to  b',  the  maximum  work  AT  +  si,  or  AT  +  SAT,  is  represented  by 
the  area  bb'iwi,  or  : 

AT  +  8AT  =  E  (T  +  8T)  In  ^ 

In  this  case  AT  +ar  >  AT,  i.e., 

SAT>0 

since  the  pressure  of  the  gas  increases  with  rise  of  temperature. 
In  some  cases,  however,  we  can  have  SAT  <  0,  as  when  the 
electromotive  force  of  a  galvanic  cell  decreases  with  rise  of  tem- 
perature (§  200). 

The  difference  SAT  is  the  amount  by  which  the  maximum  work 
of  an  isothermal  process  of  given  constant  amplitude  increases 

1  It  was  first  deduced  by  Lord  Kelvin  in  1855  (Math,  and  Phys.  Papers, 
I.,  296). 

I  2 


116  THERMODYNAMICS 

when  the  temperature  of  execution  is  raised  by  ST.     The  increase 

/3AT\  /dAT\ 

of  the  maximum  work  per  degree  is  ("^rj          =  \~jjj*)    >  tne 

suffix  indicating  that  a  constant  change  of  configuration,  or  a 
constant  amplitude,  is  maintained — the  volume  ranging  always 
between  the  two  isochores. 

This  can,  however,  be  interpreted  in  another  way,  because  £AT 
is  equal  to  the  area  ab'a'b,  i.e.,  to  the  area  of  a  circuit  including 
the  two  processes  described,  but  in  opposite  directions,  and  two 
other  connecting  processes  which  entail  no  further  expenditure 
of  work,  viz.,  the  parts  ab'  and  a'b',  in  which  the  temperature 
changes  are  effected  at  constant  configuration  by  raising  the 
temperature  of  the  medium  continuously  so  that  the  pressure  of 
the  gas  always  corresponds  with  its  temperature  as  given  by  the 
characteristic  equation 

pr  =  RT  (r  const.) 
at  every  instant. 

If  now  the  cycle  is  executed  between  the  same  isochores,  and 
with  the  same  difference  of  temperature  ST,  but  at  different 
initial  temperatures,  we  shall  have  the  lines  aa',  bb',  cc',  dd',  .  .  . 
indicating  corresponding  changes  at  different  temperatures,  and 
the  circuits  abb'a',  bcc'b',  cdd'c', . . .  indicating  corresponding  cycles 
at  different  mean  temperatures  of  execution.  With  change  of 
temperature  the  cyclic  area  moves  up  or  down  between  the  iso- 
chores, and  its  magnitude  also  changes  in  a  continuous  manner, 

the  rate  of  increase  of  area  per  1°  ascent  being  -=-f  for  the 
given  initial  temperature.  This  rate  of  increase  will  also,  in 
general,  change  with  the  temperature,  since  -~  is  also  a 
function  of  temperature. 

The  interpretation  of  the  equation  of  maximum  work, 
although  very  simple,  has  not  always  been  clearly  expressed. 


CHAPTER    V 

FLUIDS 

59.    The  Thermal  Coefficients. 

LET  us  consider  unit  mass  of  a  fluid  in  a  given  state.  Since 
the  equations  which  \ve  shall  deduce  in  this  paragraph  do  not 
depend  on  any  particular  thermonietric  scale,  we  shall  represent 
the  temperature  by  6,  where  6  may  be  the  Centigrade  tempera- 
ture, or  may  be  measured  on  any  other  temperature  scale.  The 
state  of  the  fluid  is  therefore  represented  b}T  (r,  p,  0).  If  one  of 
these  variables  increases  by  an  infinitesimal  amount  there  will, 
in  general,  be  a  corresponding  increment  in  the  value  of  each  of 
the  others,  and  there  could  be  an  infinite  number  of  corresponding 
pairs  of  values  of  the  latter  for  one  value  of  the  former.  But  if 
two  variables  are  fixed,  the  state  of  the  fluid  is  completely  defined, 
for  it  has  only  two  degrees  of  freedom  (§  26). 

The  heat  of  the  path,  8Q,  may  be  represented  by  any  one  of 
the  following  equations,  according  to  the  pair  of  independent 
variables  chosen  : 

SQ^csW  +  lJc    .         .         .         .     (1) 
=  cpd0+lltdp    ....     (2) 

=  y/0>  +  vA  ....    (3) 

The  physical  interpretations  of  the  coefficients  are  found  by 
setting  one  or  other  of  the  magnitudes  dv,  d0,  dp,  equal  to  zero. 
Thus,  if  in  (1)  we  put  dr  =  0,  d6  =  0  successively,  we  obtain  : 

Similarly  : 


Thus,  cr,  cp  are  the  amounts  of  heat  absorbed  per  unit  increase 
of  temperature  at  constant  volume  and  at  constant  pressure 
respectively.  They  are  the  specific  heats  at  constant  volume  and 
at  constant  pressure  respectively. 

/r,  /p  are  the  amounts  of  heat  absorbed  per  unit  increase  of 
volume  or  pressure  respectively,  at  constant  temperature.  They 


118  THERMODYNAMICS 

are  called  the  latent  heat  of  volume  change  (or  the  latent  heat 
of  expansion)  and  the  latent  heat  of  pressure  change  respectively. 

7,.,  7^  are  the  amounts  of  heat  absorbed  per  unit  increase  of 
volume  at  constant  pressure  and  unit  increase  of  pressure  at 
constant  volume  respectively.  They  have  received  no  special 
names. 

The  suffixes  of  the  r 's  refer  to  the  variable  maintained  constant  during  the 
change  ;  those  of  the  V&  and  -y's  to  the  independent  variable,  the  small  increase 
of  which  must  be  multiplied  by  the  corresponding  coefficient,  to  give  the  heat 
absorbed  consequent  on  the  change  of  that  variable. 

All  the  coefficients  will,  in  general,  be  functions  of  both  inde- 
pendent variables,  and  since  we  know  that  the  heat  absorbed 
depends  on  the  path  of  change,  it  follows  that  the  coefficients 
are  not,  in  general,  partial  derivatives  of  a  function  of  the  two 
independent  variables,  for  &Q  would  then  be  a  perfect  differential 
(cf.  H.  M.,  §  115). 

60.     Theorem  of  Reech. 

For  an  adiabatic  chanae  we  have : 

SQ  =  0    .        .         .         .         .     (1) 
hence  from  (1)  and  (2)  of  the  last  section : 

dv  =  -C~dd 
tfp  =  — 

£)  =C4  <a> 

ar/  Q      c^j, 
For  an  isothermal  c/mnr/c  : 

'  .         .        •.         .     (8) 


From  (2)  and  (4)  we  find  : 


'    \dre~c,. 
where  K  is  the  ratio  of  the  specific  heats. 


FLUIDS  119 

But 


9         *8 

where  eQ,  ce  are  the  adiabatic  and  isothermal  elasticities ;  hence  : 

so  that  the  ratio  of  the  adiabatic  to  the  isothermal  elasticity  of 
a  fluid  is  quite  generally  equal  to  the  ratio  of  the  specific  heat 
at  constant  pressure  to  the  specific  heat  at  constant  volume 
(Reech,  1854). 


61.     Relations  between  the  Thermal  Coefficients. 

In  the  equations  for  the  change  of  state  of  a  fluid  : 
«Q  =  c^  +  W»  \ 

dp<  .     (1) 


SQ  denotes  the  same  element  of  heat ;  hence  the  coefficients 
(e's,  Vs,  and  y's)  are  not  independent,  but  are  related.  The 
relations  are  obtained  from  the  equations  (1)  and  the  rules  for 
the  change  of  the  independent  variable  in  the  calculus.  For  the 
transformation  of  the  differentials  we  have  : 

~-  dp  -4-  ~-  dr 
dp  *    '   dr 

dv  7/1  ,  dc  , 


in  which  each  differential  is  expressed  as  a  function  of  the  other 
two  variables. 

From  equations  (2)  alone  some  useful  relations  may  be  derived. 
Thus,  if  in  the  first  we  put  d0  =  0,  i.e.,  6  is  constant,  or  the 
change  is  isothermal,  we  have  : 
"  ?)A 

rfr=°  («  const.) 


120  THERMODYNAMICS 

Similarly,  from  the  second  and  third  : 

dv 
90          d 


w 
dp 

dv\        dr  W 


0V 

If  we  put  dc  =  0,  dp  =  0,  (W  =  0  in  the  first,  second,  and 
third  equations  of  (2),  we  get 

§£-JL     ^  _  1     ^i_l 
W  ~  W  '  W  ~  W  '  dp  ~  dp    ' 

dp  dv  dv 

Turning  next  to  a  consideration  of  equations  (1),  we  observe 
that,  if  the  differentials  in  one  equation  are  arbitrary,  those  in 
the  other  two  equations  are  fixed  by  them,  and  since  each  equation 
contains  two  variables,  each  pair  of  variables  must  lead  to  four 
relations,  so  that  there  will  be  twelve  relations  for  the  three 
pairs.  These  relations  are  obtained  as  follows  : 
Let  r,  6  be  taken  as  independent  variables  : 

SQ  =  Cl,dO  +  ledv^    ....     (i.) 
Then  we  can,  from  (2),  write  the  equations 
6Q  =  Cvdd  +  lvdr 

=  7pfh>  +  7«*fr 
in  the  forms 


respectively. 

The  equations  (i.)—  (iii.)  are  now  identical,  and  by  equating 
coefficients  we  obtain  the  relations 


dp 

r,  =  %>^,etc. 

The   twelve  relations  finally   obtained   are   tabulated   below. 
They  are  clearly  not  all  independent. 


FLUIDS 


121 


Independent  variables 
0,  />. 


<,•-*+*! 


dv 


w 


From  these  equations  many  relations  between  the  c's,  Z's,  and 
7's  themselves  can  be  obtained. 

The  equations  of  §§  59 — 61  are  independent  of  the  mechanical 
theory  of  heat,  and  would  apply  equally  well  to  the  caloric 
theory.  In  the  latter  case,  however,  £Q  is  a  perfect  differential. 
They  are  also  unchanged  when  T  is  put  for  9,  where  T  is  the 
absolute  temperature.  All  the  twelve  relations  can  be  derived 
from  the  four  in  the  first  column,  together  with  the  equations  (4). 

Various  other  relations  between  the  thermal  coefficients  may 
easily  be  obtained  if  required.     Thus  : 
L 


62.     Application  of  the  First  Law  to  Fluids. 

If  we  have  unit  mass  of  a  homogeneous  fluid  having  a  uniform 
temperature  T,  and  with  its  surface  exposed  to  a  uniform  normal 
pressure  p,  its  state  can  be  defined  (if  we  regard  the  chemical 
nature  as  fixed)  in  terms  of  the  specific  volume  r,  and  the  absolute 
temperature  T. 

Let  the  state  pass  reversibly  from  (v,  T)  to  an  infinitely  near 
state  (r  +  dr,  T  +  dT).  If  SA,  BQ  denote  the  elements  of  work 
done  and  heat  absorbed, 

SA  =  pdv (1) 

SQ  =  cvdT  +  lKdv     ....     (2) 

According  to  the  First  Law,  SA  and  SQ  are  not  usually  perfect 
differentials,  but  depend  on  the  path  of  change,  whereas  their 


122  THERMODYNAMICS 

difference  is  always  a  perfect  differential,  which  was  defined  as 
the  increase  of  the  intrinsic  energy  of  the  fluid  : 

,l,i  =  SQ  -  SA      .        ...    (3) 
The  condition  that  dit  is  a  perfect  differential  requires  : 


IT      .         .     (5) 

This  shows  that  a  part  of  the  heat  absorbed  depends  on  the  change  of  tem- 
perature, and  another  on  the  change  of  volume.  The  latter  is  composed  of 
the  external  work  pdv  and  a  part  depending  on  the  change  of  intrinsic 
energy  with  volume. 

But  (§  61)  /,  =  1  (cf  —  cp)    .         .         .         .     (6) 

8T 

•        •         •     (7) 

8T 

Since  r  and  T  are  independent  variables,  the  coefficients  of 
dr  and  dT  in  (5)  and  (7)  are  identical ;  hence  : 

m  =  e*      .       •       •       •       •  (8) 


Equation  (9)  may  be  regarded  as  the  general  expression  of  the 
First  Law  as  applied  to  fluids. 

The  characteristic  equation  of  the  fluid  is  of  the  form 

p  =  F  (r,  T)      .  (10), 

and  since  this  fixes  the  external  conditions  (viz.,  the  one  that  the 
pressure  on  the  system  must  have  a  given  value)  in  order  that 
the  system  may  be  in  equilibrium,  with  chosen  values  of  the 
independent  variables  specific  volume  and  temperature,  it  may 
be  called  the  equation  of  equilibrium  of  the  fluid. 

The  two  equations  (9)  and  (10)  contain  all  that  the  First  Law 
can  teach  us  as  to  the  properties  of  the  fluid.  The  form  of  (10) 
must,  from  the  thermodynamic  standpoint,  be  regarded  as  known 
from  the  results  of  special  experiments  with  the  fluid. 

If  the  variables  (p,  T)  or  (v,  p)  are  used  instead  of  (v,  T),  another  equation 
from  the  twelve  relations  of  §  61  is  taken  instead  of  (6),  and  equation  (2)  is 
modified  so  as  to  introduce  the  chosen  variables. 


FLUIDS 


123 


If  (t>,  p}  are  taken,  the  state  of  the  fluid  is  not  in  general  uniquely 
defined.     Thus,  a  mass  of  liquid  exhibiting  a  state  of  maximum  density 

(  -V7p  =  0,  -j^  <  0  )  may  exist  in  two  states  on  opposite  sides  of  this,  having 

the  same  values  of  r  and  p,  but  different  values  of  T. 

The  equations  for  the  three  pairs  of  variables  are  given  below. 

The  deduction  of  the  equations  for  (p,  T)  and  (r,  p)  is  effected  in 
a  similar  manner  to  that  for  (r,  T),  and   s  left  to  the  reader  : 


) 


-ri- 


=   <V ;  p  + 


to 

dc  1  T 


or  ^  («  -f  F%  =  dT=  C" 

at  constant  pressure  (§  25)  ; 

or    .      du\          ,       idu 


8n 


(12) 


These  are  the  differential  equations  for  the  c's,  Z's,  and  7*8,  in 
terms  of  p,  r,  T  and  the  energy.  If  heat  is  measured  in  thermal 
units,  SQ  and  all  the  thermal  coefficients  are  to  be  multiplied 
by  the  mechanical  equivalent  of  heat,  J. 

63.     Clapeyron's  Equation. 

If  an  element  of  heat  SQ  is  added  to  unit  mass  of  a  fluid 
which   is   undergoing   a    reversible 
change,  we  have : 


If  we  suppose  the  temperature  of 
the  fluid  kept  constant,  whilst  the 
pressure  alters  in  accordance  with 
the  characteristic  equation  : 

(SQ^  =  lfdr, 
/r  being  the  latent  heat  of  expansion. 

Now  suppose  unit  mass  of  the 
fluid  taken  round  a  small  reversible 
Carnot  cycle  ABCD  (Fig.  15),  be- 
tween the  temperatures  T  and  (T  —  ST).  If  the  changes  are 


124  THERMODYNAMICS 

small,  ABCD  may  be  regarded  as  a  parallelogram,  this  amounting 
to  neglect  of  small  quantities  of  the  third  order  in  comparison 
with  the  area  ABCD  of  the  second  order.  (The  possibility  that 
opposite  sides,  <?.//.,  AB,  DC,  slope  in  opposite  directions  is 
excluded  by  the  physical  property  of  all  real  fluids,  that  the 
pressure  invariably  falls  off  with  increasing  volume.) 

Let  Sr  =  PQ  be  the  small  increase  of  volume,  and  SQ  =  l,.Sr 
the  small  amount  of  heat  absorbed,  respectively,  during  the 
change  at  T. 

The  work  done  in  the  cycle  is : 

(SA)  =  area  ABCD  =  area  FBCE  =  FB.  PQ. 

But  FB  =  increase  of  pressure  per  ST  rise  of  temperature  at 

constant  volume  = 


tfT, 

Now  (SA)  =  8Q  — ,  by  Carnot's  theorem, 


This  may  be  regarded  as  the  application  of  the  Second  Law  to 
fluids  (E.  Clapeyron,  1834). 

64.     Application  of  the  Two  Laws  to  Fluids. 

We  assume  all  the  equations  of  §  61  on  thermal  magnitudes, 
and  introduce  the  following  definitions,  each  referring  to  unit 
mass  of  the  fluid : 


=&*=?> 


fdv\ 


where  r0,  pQ  are  the  specific  volume  and  the  pressure  at  0°  C.,  V0 
is  the  specific  volume  at  1  atm.  pressure,  and  a,  ft,  f,  >/  are  the 
coefficients  of  expansion,  of  tension,  of  elasticity,  and  of  com- 
pressibility respectively. 


FLUIDS  125 

From  Clapeyron's  equation  we  have  : 

B'=ljj   .         .        .       \         .     (1), 

and  if  we  combine  this  with  the  relations 

8T 
.   dv     dv  _        dp 

lp-lvty'  ty~  ~ar 

dv 
we  find 

/,,=  -T^=-Ta'.         .         .         .     (2); 

(1)   and  (2)   enable  us  to  find   lm   lp    in    terms   of   measurable 
quantities. 

Corollary. — A  fluid  emits  or  absorbs  heat  on  isothermal  com- 
pression according  as  it  expands  or  contracts,  respectively,  with 
rise  of  temperature  at  constant  pressure. 

Thus  water  below  4°  C.  absorbs  heat  on  compression, 
above  4°  C.  emits  heat  on  compression, 
at  4°  C.  it  neither  emits  nor  absorbs  heat  on  com- 
pression. 
From  the  equation 

./  dv 
c,  =  c.+  Z,gjp 

and  Clapeyron's  equation  we  find  : 

c,,-C,=  T^.||  =  Ta',3'    .         .         .     (3) 

But  =-. /,'  =  =: 


dp 

which  give  the  difference  of  the  specific  heats  in  terms  of  the 
coefficient  of  expansion  and  the  coefficient  of  tension  or  elasticity. 

Example. — In  the  case  of  mercury  : 

Cp  =  0-0333 
T  =  273 


126  THERMODYNAMICS 

r/>  _          1013250    dyne 
V  dv  ~        0-0000039  ~^~ 
v  =  1/13-596 

^  =  0-0001812«. 
31 
To  obtain  ce  in  calories  we  must  divide  by  J  : 

_  r   =     273  X  1013250  X  (Q-Q001812)2 

C"       ("~  0-0000039  X  13-596  X  4'19  X  107 

.-.  cv  =  0-0292,  cljco  =1-1. 


We  also  readily  find  : 

c,,  —  c,  =  Tr0(ea)a    .          .          .          .     (4) 
/„=  TaV  =  Tae      .          .          .       .   .     (5) 
In  the  application  of  the  First  Law  we  have  : 

(hi  -  SQ  —  SA  =  crdT  +  (/„  -  2))  tie 
du  m  dp 

.-.  ^  =  i,-P=i^-P 

du 

BT  =  r" 

9/        ^> 


_ 

~  da 
The  condition  that  </«  is  a  perfect  differential  : 


leads  to  the  relation 

dc,.  _  dl,,       dp  _  ,„  d2p  ,R, 

d~v  ~8T~8T~      ^F 
Similarly  we  can  deduce  : 


Finally,  consider  the  case  of  adiabatic  compression.    Taking  T,  _ 
as  independent  variables,  we  have 

SQ  =  <y/T  +  /^. 
In  an  adiabatic  change  SQ  =  0, 


-~7;-a^T    •     •     •  (8) 

For  convenience,  put  (S^)Q  =  a>,  (ST)Q  =  d, 


FLUIDS  127 

then 

3  =  ^o>         '.         .'      .         .     (9) 
e& 

where  •&  =  rise  of  temperature  consequent  on  an  adiabatic 
increase  of  pressure  <« ;  p0  —  density  at  0°  C. ;  a  =  coefficient 
of  expansion. 

Exercises.— (1}  If  5v  is  the  diminution  of  volume  producing  o>,  show  that 

A  =  T  —  5i>  = ™ 80    .  .     (10). 

ce  r,,  -  T«6.v0« 

Equations  (8)— (10)  were  deduced  by  Lord  Kelvin  (1857),  and 
verified  by  Joule  (1859).  In  the  case  of  water  : 

d  <  0  if  T  <  (273  +  4°  C.),  since  a  <  0, 
and  S  >  0  if  T  >  (273  +  4°  C.),  since  a  >  0. 

(2)  Prove  that : 

(°!L\    =  T>JL 
\ZvJT          31. 

(3)  Show  that  _p/T  —  f(v)  is   the   characteristic  equation  of   a  fluid   the 
intrinsic  energy  of  which  is  independent  of  the  volume. 

65.     Relation  between  Isotherms  and  Adiabatics. 

There  is  a  perfectly  general  theorem,  which  applies  to  all 
bodies,  or  systems  of  bodies,  to  the  effect  that  an  adiabatic,  when 
it  crosses  an  isotherm  on  the  indicator  diagram,  is  more  inclined  to 
the  v-axis,  or  the  adiabatic  is  steeper  than  the  isotherm. 

We  shall  not  attempt  any  general  proof  of  the  theorem  at  this 
point,  but  a  few  illustrations  may  be  given. 

If  a  fluid  is  enclosed  in  a  cylinder  with  walls  impervious  to  heat,  and 
compressed,  it  is  either  heated  or  cooled  according  as  a,  the  coefficient  of 
expansion ,  is  >  0  or  <  0.  In  both  cases  the  volume  would  have  increased , 
and  the  pressure  risen  more  rapidly  than  if  heat  were  allowed  to  pass  out,  or 
in,  through  the  walls.  Thus,  for  a  given  diminution  of  volume,  the  rise  of 
pressure  is  greater  under  adiabatic  than  under  isothermal  conditions. 

The  theorem  also  applies  to  a  heterogeneous  system,  such  as  a  liquid  in 
presence  of  its  saturated  vapour,  or  in  presence  of  the  solid.  In  the  former 
case,  vapour  is  liquefied  by  compression  and  gives  out  its  latent  heat.  Under 
isothermal  conditions  this  would  escape  as  fast  as  produced,  but  if  the  heat  is 
compelled  to  remain  in  the  system,  it  raises  the  temperature  and  thereby 
increases  the  pressure.  If,  on  the  other  hand,  a  mixture  of  ice  and  water  is 
compressed,  ice  melts  and  the  mass  is  cooled  by  abstraction  of  heat.  If  heat 
is  allowed  to  enter  from  outside,  so  as  to  restore  the  original  temperature, 
more  ice  melts,  and  the  pressure  falls  by  reason  of  the  contraction. 


128 


THERMODYNAMICS 


The  theorem  under  discussion  is  a  particular  case  of  a  very 
general  principle,  which  was  stated  by  Maxwell  (1871)  in  the 
form  that  "  a  force  producing  alteration  of  the  state  of  a  con- 
strained system  is  always  greater  than  a  similar  force  producing 
the  same  alteration  in  an  unconstrained  system." 

Two  isotherms,  isochores,  adiabatics,  or  generally  any  two 
thermal  lines  of  the  same  kind,  never  cut  each  other  in  a  surface 
representing  the  states  of  a  fluid  with  respect  to  the 


three  variables  of  the  characteristic  equation  taken  as  co-ordi- 
nates, for  a  point  of  intersection  would  imply  that  two  identical 
states  had  some  property  in  a  different  degree  (<?.//.,  two  different 
pressures,  or  temperatures).  Two  such  curves  may,  however, 


FIG.  16. 

cross  each  other  on  a  plane  diagram  representing  a  projection  of 
the  lines  in  the  space  model,  as  was  pointed  out  by  Eiicker(1874). 
This  may  be  illustrated  by  considering  the  case  of  water. 

Consider  a  mass  of  water,  and  let  A,  B,  on  the  isopiestic  AB  (Fig.  17)  repre- 
sent the  pressure  and  specific  volume  at  0°  and  at  4°  respectively.  If  we  start 
with  the  water  in  any  state  intermediate  between  these,  say  P,  it  will  contract 
on  heating  till  it  reaches  B,  the  point  of  maximum  density.  Further  heating 
causes  it  to  repass  through  the  same  states  of  density,  but  at  different 
temperatures.  It  reaches  P  having  an  excess  of  heat  equal  to  that  supplied, 
the  work  of  compression  and  of  expansion  being  equal  and  opposite.  Hence 
every  point  between  A  and  B  may  be  regarded  as  lying  on  two  intersecting 
adiabatics.  If  the  water  is  cooled  at  A  till  it  freezes  it  passes  to  the  right 
of  A,  but  if  heat  is  still  further  abstracted  from  the  ice  it  contracts,  and 
passes  again  through  all  points  between  A  and  B.  Thus  three  adiabatics 
pass  through  every  one  of  these  points,  each  corresponding  to  a  different 
state.  The  lines  do  not  really  cross  in  space  ;  the  apparent  intersections  are 
due  to  the  fact  that  part  of  the  diagram  overlies  another,  like  the  contour 
lines  of  an  overhanging  cliff. 


FLUIDS  1*29 

66.     Massieu's  Theorems. 

Let  V  be  the  free  energy  of  unit  mass  of  a  fluid,  and  let  ^  be 
known  as  a  function  of  r  and  T  :  tKr,T).  Then  it  is  easily  shown 
from  the  results  of  §  59  and  the  present  chapter  that : 


WKr 
~W 


_  _  y^y 
dice 


Thus,  if  one  has  determined  ^  as  a  function  of  r  and  T,  all  the 
thermodynamic  properties  of  a  fluid  can  be  caressed  in  terms  of  the 
free  energy,  ^(r,T). 

Again,  if  <f>(p,T),  the  potential  of  unit  mass  of  a  fluid,  is  known 
as  a  function  of  p  and  T,  we  have  : 


T(**-\ 

^-T-^-T^  +  %^- 

dp* 


_  _ 

^  -  a$  '     •  a^  '  '"  ~      A  ar?^' 

dp  dp 

Thus,  all  the  ihermodynamic  properties  of  a  fluid  can  be  expressed 
in  terms  of  the  potential  ^(_p,T). 

Thus,  all  the  thermodynamic  properties  of  a  fluid  are  known  if 
we  are  in  possession  of  a  single  function  of  the  independent 
variables  in  terms  of  which  the  state  is  denned. 

If  these  variables  are  r,  T  the  function  is  the  free  energy  i/r ; 
if  they  are  p,  T,  the  function  is  the  potential  <j>. 


180  THERMODYNAMICS 

This  theorem,  which  is  of  great  importance,  was  established  by 
F.  Massieu  in  1869  (Journ.  de  Phys.,  6,  216,  1869  ;  C.R.  69,  858, 
057,  1869).  It  may  be  extended  to  all  systems  by  an  application 
of  the  methods  employed  in  §§  58,  59  (cf.  Duhem,  Traite  de 
Mecanique  chimique,  L,  pp.  104 — 113). 


CHAPTER  VI 

IDEAL   AND    PERMANENT    GASES 

67.     The  Characteristic  Equation  of  a  Gas. 

The  state  of  unit  mass  of  a  gas,  like  that  of  any  other  fluid,  is 
denned  by  any  pair  of  the  variables  p,  r,  0,  and  its  characteristic 
equation  is  therefore : 

/Uv,0)  =  0  .      (1) 

If  the  temperature  is  maintained  constant,  the  volume  is  found 
experimentally  to  be  approximately  inversely  proportional  to  the 
pressure  (Boyle's  law) : 

pv  =  <t>(0)          ....     (2) 

If  the  pressure  is  maintained  constant,  the  volume  is  found  to 
increase  by  approximately  the  same  fraction  of  the  volume  at 
0°  C.  for  each  degree  rise  of  temperature  (law  of  Dalton  and  Gay- 
Lussac) : 

r  =  r0(l  +  o0)  .         .         .     (3), 

where  a  =  —  (^J  =  coefficient  of  expansion,     a  is  very  nearly 

the  same  for  all  gases  :  a  =  1/273. 

The  change  which  the  volume  of  the  gas  undergoes  during  any 
simultaneous  change  of  p  and  6  is  the  sum  of  the  changes  it 
undergoes  when  p  and  6  are  separately  altered,  since  these  are 
independent  variables. 

For  the  change  of  p  from  pi  to  p%  at  #1 : 
p&'  —  pii\  =  0 

For  the  change  of  6  from  0V  to  02  at  p* : 

ra  _  1  +  afli 

r'       1  +  o0i 

.-.  pin  (1  +  a02)  =  pars  (1  +  a0i) 

Put  0i  =  Q     .'.  &  =  po,  n  =  r0,  0-2  =  0, 

then  pv  =^or0(l  +  a0)    .         .         .         .     (4) 

K   2 


132  THERMODYNAMICS 

We   shall   now   define   the   absolute  yas   temperature   by   the 
equation 

.     (5) 

,     (6) 


(T)  =  °  +  a 


Corollaries.— (I)  If  £  =  -  ( £}    is  the  coefficient  of  tension; 

7)   \utj  J  v 

and        p  =  2)0  (1  +  aO).. 

(2)  The  absolute  gas  temperatures  are  proportional  to  the 
volumes  at  constant  pressure,  and  to  the  pressures  at  constant 
volume : 


The  isotherms  of  a  gas  have  the  equation  (2),  and  form  a  series 


FIG.  18. 


FIG.  19. 


of  rectangular  hyperbolas  ;  the  isopiestics  (p  const.)  and  isochores 
(v  const.)  have  the  equations  : 


and  form  series  of  straight  lines  radiating  from  the  zero  point 
(T)  =  0,  i.e.,  0=  —I/a  (Figs.  18  and  19). 

If  we  put  poi-tfi  =  i-       .         .         .         .         .     (7) 

we  may  write  the  general  gas  equation  (6)  in  the  form  : 

pr  =  r(T)  ....     (8) 

This  is  the  characteristic  equation  of  a  gas  (Clapeyron,  1834), 


IDEAL   AND  PERMANENT   GASES  133 

and  for  the  permanent  gases,  such  as  nitrogen,  oxygen,  hydrogen, 
etc.,  we  can  put : 

/•  —  MO  (9) 

~273   ' 

r  is  called  the  gas  constant. 

If  instead  of  unit  mass  we  consider  a  mass  in,  of  volume  V0 
at  pQ,  00  • 

j>V  =  mr(T)  =  r'{T)     .        .        .        (10) 

For  equal  masses  of  different  gases,  the  constants  ;•'  are 
inversely  proportional  to  the  densities. 

It  is  very  important  to  observe  that,  inasmuch  as  a  is  the  same 
for  all  gases,  these  have  characteristic  equations  of  the  same 
form  (B.  P.  E.  Clapeyron,  Jouni.  de  I'Ecole  Polytechn.,  18,  170, 
1834;  Pogg.  Ann.,  59,  446,  1843). 

68.     Molecular  Weight. 

Definitions. — (1)  The  absolute  density  of  a  gas  (S)  is  the  weight 
of  1  litre  in  grams  at  0°  C.  and  1  standard  atmosphere  pressure. 
(The  weight  is  reduced  to  sea-level  and  latitude  45°.) 

If  u-  =  weight  of  1  litre  at  0°  C.  and  p  atm. 

8  =  «>x|x(l  +  «0).        •        •        •    (1) 

(2)  The  normal  density  of  a  gas  (D)  is  32  times  the  ratio  of  the 
weights  of  equal  volumes  of  the  gas  and  of  oxygen  under  normal 
conditions  (1  atm.,  0°  C.). 

D  =  32~.  =  32^          .         .         .     (2) 

\V  \ 

If  the  general  gas  law  held  rigorously  true  for  all  gases,  this 
ratio  would  be  independent  of  temperature  and  pressure,  by  (1), 
and  D  would  depend  only  on  the  chemical  composition  of  the  gas. 

(3)  The  molecular  weight  (M)  of  a  gas  is  the  normal  density, 
corrected  if  necessary  for  the  deviations,  exhibited  by  the  gas, 
from  the  gas  laws. 

Since  the  permanent  gases  deviate  only  slightly  from  these 
laws,  we  shall  at  present  regard  them  as  ideal  gases  ;  hence : 

M  =  D (3). 

The  slight  deviations  may  be  taken  into  account  by  a  method 
described  in  §  80. 

The  definition  of  molecular  weight  just  given  leads  at  once  to 
the  conclusion  that  equal  volumes  of  different  permanent  gases, 


184  THERMODYNAMICS 

under  the  same  conditions  of  temperature  and  pressure,  contain  the 
same  number  of  molecular  weights. 

This  is  called  Arogadro's  theorem  (1811)  ;  it  appears  here 
simply  as  a  definition  of  molecular  weight,  and  this  is  really  the 
manner  in  which  the  relation  is  applied  in  chemistry.  The 
kinetic  theory  of  gases  gives  a  new,  and  much  deeper,  signifi- 
cance to  the  statement  by  introducing  the  conception  of  the 
molecule  ;  this,  however,  does  not  concern  us  in  thermodynamics, 
and  since  the  molecular  weights  are  purely  relative  numbers,  the 
deductions  made  in  this  book  are  equally  strict  whichever  stand- 
point is  adopted. 

The  molecular  weight  of  a  substance  is,  for  a  large  number  of 
calculations,  far  more  convenient  than  unit  mass.  This  depends 
on  the  fact  that  a  large  number  of  properties  are  independent  of 
the  nature  of  the  substance,  and  depend  only  on  the  number  of 
molecular  weights  present  ("  molar,"  or  "  colligative,"  properties) 
(cf.  Chap.  XI.  on  "  Solutions  "). 

69.     The  Molar  Gas  Constant. 

It  is  evident  from  the  gas  equation 

that  >•'  has  the  same  value  for  all  gases  if  quantities  are  selected 
which  have  the  same  pressure,  volume,  and  temperature.    Thus  r' 
has  the  same  value  for  a  molecular  weight  of  any  ideal  gas.     If 
we  denote  this  value  by  R,  and  if  r  is  the  value  for  unit  mass, 

R  =  Mr    .         .         .         .    .     .     (2) 

/.  2w==fi(T)         .         ."       .         .    (3), 

where  r,  as  before,  denotes  the  specific  volume. 

If  we  put  M.V  =  v 

for  the  molecular  volume  of  the  gas  at  0>,T) : 

If  we  consider  any  mass  in  of  the  gas,  occupying  a  total 
volume  V : 

/  =  -(T) 

Vfl. 

•         .         -     15), 


IDEAL   AND   PEEMANENT   GASES  135 

where  n  =  wi/M  is  the  number  of  gram-molecules,  or  mols,  of  the 
gas  in  the  volume  V.  (Goldberg,  1867X 

Equation  (4)  applies  to  a  mol.  of  any  gas,  R  having  a  constant 
value  which  can  be  calculated  as  soon  as  we  know  the  specific 
volume  of  the  gas  at  a  given  temperature  and  pressure,  and  its 
molecular  weight. 

At  atmospheric  pressure  and  at  the  temperature  of  melting  ice,  32  gr.  of 
oxygen  occupy  a  volume  of  22,412  c.c.  (corrected  for  a  slight  deviation  from 
the  gas  laws)  : 

.-.  p  =  l,  (T)  =  273-09  (this  is  more  accurate  than  the  value  273  previously 
used),  f  =  22,412 

.-.  E  =  1  X  22'412  =  0-08207     L  atm-  . 
273-09  degree  C. 


=  0-08207  X  1013-13  X  10J  =  8'315  X  10'  =--7- 

degree  C. 

=  0-08207  X  1033200  =  84795 

=  0-08207  X  24-191  =  1'9854 

Guldberg,  Forhandl.  Videnskabs.  Selskabet,  Christiania,  Oct.,  1867, 
Ostwald's  Klassiker,  139,  6;  A.  Horstmann,  Berl.  Ber.  U,  1243,  1881; 
J.  H.  van't  Hoff  ,  Klassiker,  No.  110,  30  ;  D.  Berthelot,  Zeitschr.  ElektrocJtem. 
10,  621,  1904;  W.  Nernst,  ibid.,  629. 

70.     Ideal  Gases. 

The  word  "  approximately  "  has  been  used  in  framing  the  laws 
of  gases,  because  no  actual  gas  exists  which  obeys  these  laws 
strictly.  By  "  strictly  "  we  refer  to  that  closeness  of  agreement, 
within  the  limits  of  experimental  error,  observed  in  such  laws  as 
Newton's  law  of  gravitational  attraction,  the  law  of  conservation 
of  mass,  the  law  of  conservation  of  energy,  and  Faraday's  laws 
of  electrolysis.  The  degree  of  approximation  is,  however,  different 
for  different  gases,  the  so-called  "  permanent  gases"  approaching 
very  closely  to  exact  agreement,  whereas  easily  liquefiable  gases  like 
sulphur  dioxide,  carbon  dioxide,  and  ammonia  deviate  markedly 
from  the  gas  laws.  It  has  been  found,  however,  that  in  all  cases 
the  deviations  become  smaller  and  smaller  as  the  pressure  is 
reduced  more  and  more,  until  under  very  small  pressures  all 
gases  approach  an  ideal  limiting  state,  which  would  be  attained 
when  the  pressure  has  become  infinitely  small.  Such  a  limiting 
form  of  the  gaseous  state  we  shall  call  an  Ideal,  or  a  Perfect,  gas. 

For  the  purposes  of  theoretical  reasoning  we  shall  further 


136  THERMODYNAMICS 

suppose  that  such  an  ideal  gas  can  exist  under  ordinary  con- 
ditions of  temperature  and  pressure.  This  is  very  nearly  realised 
by  hydrogen  and  helium,  and  moderately  closely  by  the  other 
permanent  gases. 

It  would  appear  at  first  sight  necessary  to  define  an  ideal  gas 
as  one  which  strictly  obeys  all  the  gas  laws.  As  a  matter  of  fact 
we  can  prove  that  if  it  conforms  to  tico  conditions  it  will  conform 
to  all  the  conditions  we  shall  take  as  defining  an  ideal  gas. 

Definition. — An  ideal  gas  is  a  fluid  which  obeys  Boyle's  law, 
and  the  internal  energy  of  which  is  independent  of  the  volume  : 

pr  =  const.) 

/dU\   _  ~         ~  T  constant. 
\dr/T~  I 

The  latter  condition  follows  from  a  very  important  result 
established  experimentally  by  Joule  (J.  P.  Joule,  Phil.  Mag.  [3] , 
23,  343,  435  (1843) ;  26,  369, 1845). 

71.     Theorem  of  Joule. 

The  isothermal  expansion  of  an  ideal  gas  is  an  aschistic  process. — 
If  a  mass  of  gas  expands  isothermally,  the  heat  absorbed  is  equal 
to  the  external  work  done. 

This  result  could  be  inferred  from  the  agreement  between 
the  mechanical  equivalent  of  heat  calculated  by  J.  B.  Mayer 
(1842)  and  that  determined  experimentally  by  Joule  (1843). 

If  1  gr.  air  is  warmed  through  1°  and  at  the  same  time 
allowed  to  expand  under  atmospheric  pressure,  0'2408  cal.  of  heat 
are  absorbed  i.e.,  cp  =  0'2408  cal.  But  if  the  heating  is  carried 
out  at  constant  volume,  only  01713  cal.  are  absorbed,  i.e.,  ce  = 
01713  cal.  If  we  assume,  as  was  tacitly  done  by  Mayer,  that 
the  difference, 

<-,,  -  ce  =  0-0695  cal., 

is  entirely  spent  in  doing  the  external  work  of  expansion  against 
the  atmospheric  pressure,  which  amounts  to  the  theorem  just 
stated,  we  have : 

(cp  —  cr)  X  J  =  external  work. 

=  pressure  X  increase  of  volume. 
1    gr.  air  at  0°  C.    has   a  volume    773-4  c.c.  /.    expansion  = 

773-4  X  ,-=_  =  2-83  c.c.,  and  1  atm.  pressure  =  1033    -^ 


IDEAL   AND  PERMANENT   GASES 


137 


.-.  external  work  =  1033  X  2'83  =  2933'4  gr.  cm. 

.^  eai.  =          =  42,060  g,  cm.  =J. 


The  close  agreement  of  this  with  the  direct  value  (41,880) 
verifies  the  theorem. 

The  assumption  tacitly  introduced  by  Mayer  was  called  into 
question  by  Joule,  who  pointed  out  that  it  certainly  is  not  true  if 
the  expanding  substance  is  a  liquid,  and  who  in  1845  subjected 
"Mayer's  hypothesis  "  to  the  test  of  a  direct  experiment.  The 
principle  was  the  following  : 

If  a  volume  of  air  (or  other  gas)  is  allowed  to  expand  without 
doing  external  work  the  process  is  adynamic  : 


TBS 


If,  therefore,  the  process  is  aschistic,  i.e.,  SA  =  SQ, 
there  ought  on  the  whole  to  be  no  heat  absorbed  from  or  given  up 
to  the  surroundings  : 

/.  SQ  =  0. 

Two  copper  globes,  A  and  B,  the  first  containing  air  at  22  atm. 
pressure,  and  the  second  vacuous,  were  immersed  in  a  can  of 
water.  On  opening  the  tap 
connecting  the  globes,  the  expan- 
sion occurred,  but  after  the 
water  had  been  stirred  no  change 
of  temperature  was  observed. 
Joule  therefore  concluded  that  : 
"  no  change  of  temperature 
occurs  when  air  is  allowed  to 
expand  without  developing  mechanical  power  [i.e.,  doing  external 
work]  ."  The  globes  A  and  B  and  the  tap  C  were  then  placed  in 
separate  cans  of  water  (Fig.  20)  and  the  experiment  repeated. 
A  fall  of  0'595°  C.  per  kilogram  of  water  occurred  in  A,  a  rise  of 
0'606°  C.  per  kilogram  in  B,  and  a  rise  of  0*078°  C.  per  kilogram 
in  C.  Thus,  within  the  limits  of  error  : 

(i.)  The  same  amount  of  heat  is  lost  by  the  gas  in  A  com- 
pressing that  in  B  as  is  produced  in  B  by  the  compression,  the 
total  change  being  zero. 

(ii.)  No  heat  is  absorbed  or  evolved  in  the  tap. 

Since,  from  the  conditions  of  the  experiment,  it  was  found  that 
the  thermometer  used  would  not  have  been  affected  if  a  change 


FIG.  20. 


138  THERMODYNAMICS 

of  temperature  of  less  than  1'88°  C.  had  occurred  in  the  air,  Lord 
Kelvin  in  1851  suggested  modifications  in  the  method  with  a  view 
to  increasing  its  sensitiveness,  and  in  1852  he  and  Joule  carried  out 
the  classical  "  Porous  Plug  experiment,"  forcing  the  gas  through 
a  plug  of  silk,  and  measuring  its  temperature  before  and  after 
passage.  The  experiments,  described  in  detail  later,  showed  that 
—  contrary  to  Mayer's  hypothesis  —  there  was  a  very  slight  cooling 
effect  with  air  and  carbon  dioxide,  and  a  very  slight  warming  effect 
with  hydrogen.  Still,  for  the  permanent  gases  we  may  take  the 
result  of  Joule's  original  experiment  as  being  very  nearly  true. 

Thus  (SQ  -  2A)T  =  Ua  -  Ui  =  0 

or  the  intrinsic  energy  of  an  ideal  gas  is  independent  of  the 
volume  : 


The  fact  the  changes  of  temperature  occur  during  the  expansion  does  not 
affect  the  argument,  since,  as  we  are  dealing  with  the  First  Law  only,  it  is 
the  initial  and  final  states  alone  which  are  of  account. 


72.     Difference  of  the  Specific  Heats  of  Air. 

The  calculation  of  Mayer  was  thrown  into  a  different  form  by 
Rarjkine  (1850),  who  showed  that,  instead  of  estimating  the 
mechanical  equivalent  of  heat  from  the  difference  of  the  specific 
heats  of  air,  one  could  take  Joule's  value  of  the  mechanical 
equivalent  and  the  known  ratio  of  the  specific  heats,  and  thence 
determine  the  specific  heats  themselves. 

J  (<.,  -  c,)  =  p(r,  -  n)  =  r  [((T)  +  1}  -  T]  =  r. 
r  =  2933-39,  J  =  41,880  gr.  cms. 
/.  41,880  (cp  -  cv)  =  2933-39. 

/.  41,880        -l    =§§?*?§. 


Again,  cjc,,  =  1*405,  from  experiment. 
.-.  ce  =  0-172  ;  cp  =  0-2417. 

These  values  differed  from  the  results  of  Delaroche  and  Berard, 
available  at  the  time,  but  were  afterwards  confirmed  by  the  more 
accurate  work  of  Regnault  (Regnault,  Mem.  de  I'Inst.  France, 
26,  1862). 


IDEAL   AND  PERMANENT   GASES  189 

73.     Equations  for  Ideal  Gases. 

The  equation  of  §  66  : 

'  (1) 


applies  to  unit  mass  of  any  homogeneous  fluid. 
If  the  change  is  isothermal  : 

dT  =  0    .         .         .         .         .     (2) 

•••*=(*+£),* 

l,  =  P~          •        •        •    (8) 


For  an  ideal  gas,  however, 

oule's  theorem)       .         .         . 

(5)  ; 


•=-  =  0  (Joule's  theorem)       .         .         .     (4) 


/.  with  Clapeyron's  equation  (§63)  we  get : 

—    v  __  1  //?\  . 

~  T  "~  T     '        '        '         •  W ' 
/.by  integration : 

/  (v)  taking  the  place  of  the  integration  constant,  since  we  are 
dealing  with  partial  differentials. 

Thus  p  =  T  X  const,  (r  constant)    .         .         .     (8) 
But  pv  =  const.  (T  constant)         .         .         .     (9) 
/.  pv  =  const.  X  T. 

or  pv  =  rT (10) 

which  is  the  general  gas  equation. 
We  now  take  the  equations 

',=  «-!   ....  (ID 


.         .         .         .     (12) 

\J    J. 

of  §§  61,  64. 

From  (9)  we  get  cf  = ' 

from  (5). 


140  THERMODYNAMICS 

Thence  from  (12),  at  the  temperature  of  melting  ice  : 


or  the  coefficient  of  expansion  of  an  ideal  gas  is  equal  to  the  reciprocal 
of  the  absolute  temperature  of  melting  ice. 

But  the  gas  temperature  (T)  has  been  denned  as 

(T)  =  0+1  =  0  +  To      .        .         .     (15), 
a 

and  since  T  =  6  +  T0  •  .        .         .     (16) 

the  size  of  the  absolute  degree  being,  by  convention  equal  to  that 
of  the  Centigrade  degree,  we  have 

(T)=  T          ..  \   ...     .r..r.,    .     (17), 

so  that  absolute  thermodynamic  temperatures  are  equal  to  the  gas 
temperatures  measured  with  an  ideal  gas  thermometer. 

Relation  (17)  is  an  equality,  but  not  an  identity  ;  it  is  true  only 
for  a  perfect  gas. 

The  thermometer  may  operate  either  at  constant  volume,  or  at 
constant  pressure,  since,  if  the  gas  obeys  Boyle's  law,  /3  =  a. 

From  the  general  equation  of  §  62  : 

SQ  =  cVfT  +  7A  .....  (18) 

and  (5)  it  follows  that  the  expression  of  the  first  law  for   ideal 
gases  may  be  written  in  the  form 

5Q  =  cvdT  +  pclr  ....     (19) 

Corollary  1.  —  The  specific  heat  at  constant  volume  of  an  ideal 

gas  is  a  function  of  temperature   alone.     For  cv  =  («K)  ,  and 

\o  -L  /  v 

u  is  independent  of  v. 
By  means  of  the  general  gas  law 

pv  =  rT       '    .      '  .    •    .         .     (20) 
or  ^r  =  RT          v'        .""  '%         .   (20a) 
we  can  eliminate  p,  r,  or  T  from  (19). 

If  equation  (20)  is  used,  all  the  thermal  magnitudes  refer  to 
unit  mass,  i.e.,  ct>,  cf  ;  if  (20a)  is  used  they  refer  to  a  mol,  i.e., 
Cp,  CM  where  C,  =  Mcv,  etc. 
(a)  Elimination  of  p  : 

bQ  =  cvdT+1^dv         .        .         .     (21) 

V 

or  8Q'  =  C//T  +  —  ifo         .         .         .  (21rt) 


IDEAL   AND  PERMANENT   GASES  141 

(b)  Elimination  of  r  : 

r=rT/p.     .:dv=ldT-^dp 

.-.  8Q  =  (Cr  +  /•)  dT  -  jdp .      '  . .       .      (22) 

TIT 
Similarly  6Q'  =  (C,  +  R)  dT dp        .        .  (22a) 

Corollary    2. —  <•„  =  ec  +  r      .         .        .         .    (23) 

or  C,  -  Cr  =  R  =  1-9854  g.  cal.        .         .  (23a) 
The  difference  of  the  specific  heats  referred  to  unit  volume  is 
the  same  for  all  gases  (Clapeyron,  1834). 
Corollary  3. — cp  is  a  function  of  T  alone. 

(c)  Elimination  o/'T: 
Differentiate  (20) 


.-.  6Q  =  —  rdp  -f  ^-^— pdv  .         .         .     (24) 

or  8Q'  =  ^  rdp  -\ ^ — '  pdv         .         .  (24«) 

Exercise. — Show  that,  for  a  small  change  of  state  of  an  ideal 
gas: 

5Q  =  c^T  +  (cp  -  cv)  - 

I 

(25) 


74.     Integration  of  the  Equations  for  Ideal  Gases. 

(1)  Isothermal  Ej.yansion  : 

dT  =  0, 

.-.  bQ=pdv=:'^-dv 


•>:?—*•- 


il  =  rT/»  =  .         •         •     (1) 
Pi 

•I    r, 


where  "  In  "  denotes  the  natural  logarithm. 


142  THERMODYNAMICS 

Corollary  1.  —  If  an  ideal  gas  changes  its  volume  reversibly 
without  alteration  of  temperature,  the  quantities  of  heat  absorbed 
or  emitted  form  an  arithmetical  progression  whilst  the  volumes 
form  a  geometrical  progression  (Sadi  Carnot,  1824). 

E.g.,v  =  1         2        4        8         16    .         .        .        , 
Q  =  0         1        2         3  4     .         .         .-        . 

Corollary  2.  —  Since  P  pdv  =  [pv^  —     2  vdp 
Jvi  jpi 

and  [pvf  =  j>2r2  —  p\v\  =  0  (Boyle's  law), 

£i       .         .    •     .         .     (11) 
P* 

J-1      .         ...     (Ic) 

Corollary  3.  —  All  gases,  if  equal  volumes  of  them  are  taken 
under  the  same  conditions  of  temperature  and  pressure,  evolve  or 
absorb  equal  quantities  of  heat  if  compressed  or  expanded  by  equal 
fractions  of  their  volumes  (Dulong's  rule). 

Corollary  4.  —  If  (p\,  rlf  TI)  and  (p2,  r2,  TI)  are  the  initial  and 
final  states  (T  constant) 


(2)  Adiabatic  Expansion  : 

SQ  =  0 

.-.  ct,dT  +  (cp-Cl)~ 
Divide  by  ccT,  and  put  c,,/c,,  =  K  : 


In  the  integration  of  (2)  we  must  know  K  as  a  function  of  v  and 
T.  Now  it  has  been  shown  that  cpt  cv  are  functions  of  T  only, 
hence  K  is  a  function  of  T  alone.  We  shall  now  assume,  as  an 
experimental  fact,  that  cv  is  independent  of  temperature  in  the  case 
of  permanent  gases.  This  was  verified  to  a  close  degree  of 
approximation  by  Regnault  (cf.  §  7). 

Corollary  l.—c,,  =  CK  +  r  is  independent  of  temperature. 

Corollary  %,—cJc,.  =  K  is  independent  of  temperature. 

Corollary  3.  —  The  internal  energy  of  an  ideal  gas,  the  specific 


IDEAL   AND   PERMANENT   GASES  143 

heat  of  which  is  independent  of  temperature,  is  represented  by  an 
expression  of  the  form 

u  =  MO  +  crT, 

where  «0  refers  to  absolute  zero. 

We  may  now  integrate  equation  (2)  on  the  assumption  that  cr, 
and  therefore  K,  are  constant : 

InT  +  (*  —  !)  Inv  =  constant 
.-.  Tt"-1  =  constant, 

-  <"> 

If  we  eliminate  r  or  T  from  (2a)  by  means  of  the  equation 
pv  =  rT,  we  obtain 


Pi        \<V 

Equation  (2c)  may  be  written  in  the  form 
pv*  =  constant 

which  is  the  well-known  equation  for  adiabatic  expansion  of  a 
permanent  gas,  deduced  before  the  time  of  the  mechanical  theory 
of  heat  by  Laplace  and  Poisson.  The  present  deduction  is  due 
to  Clausius  (1850). 

Table  of  Values  of  K. 

Air  1-403  Hydrogen  1'4084 

Oxygen  1*398  Carbon  dioxide  1'2995  (15°) 

1-280  (100°) 

Steam  T305  (100°)  Ether  vapour  1-097  (20°) 

Nitrogen  1'400  Carbon  monoxide  T403  (0°) 

1-397  (100°) 

Chlorine  T323  (360°)  Iodine  1'307  (360°) 

Ammonia  1'317  (0°)  Chloroform  I'llO 

Mercury  T674  (300°)  Argon  T667 

The  highest  value  of  K,  1'667,  which  is  that  predicted  by  the  kinetic  theory 
of  gases,  is  observed  only  with  monatomic  gases  (argon,  mercury).  Diatomic 
gases  have  the  value  1-4,  triatomic  1-3,  and  K  decreases  with  increasing 
molecular  complexity  (cf.  Chap.  XVIII.). 

The  cooling  produced  when  a  strongly  compressed  gas  is  allowed  to 


144 


THERMODYNAMICS 


expand  adiabatically,  doing  external  work  against  the  confining  pressure, 
was  utilised  by  Cailletet  to  bring  about  the  liquefaction  of  the  so-called 
"permanent  gases"  (02>  N2,  H2,  CO,  CH4,  NO),  which  had  resisted  the 
efforts  of  Natterer  (1844),  who  exposed  them  to  pressure  alone. 

75.    Determination  of  the  Ratio  of  Specific  Heats  of  a  Gas. 

(1)  General  Case. — Consider  a  mass  m  of  a  gas  enclosed  in  a 

cylinder  fitted  with  a  piston  and  in  connexion  with 

I  a   manometer    capable   of   indicating    very    rapid 

changes    of   pressure  (Fig.   21).      Let  po  be   the 

initial  pressure. 

•••••^  By    rapidly    withdrawing    the    piston,    let    the 

volume  be  increased  by  AV ;  the  increase  of 
specific  volume  is  AT  =  AV/wt.  The  indicated 
pressure  will  be,  the  instant  after  the  expansion, 
less  than  p0 ;  let  it  be  p^ 

Pi  —  Po  =  Ap. 

Since  the  change  is  rapid,  it  may  be  assumed 
FIG.  21.        to  be  adiabatic, 


•     (1), 


where  a  is  a  small  magnitude  which  tends  to  zero  when  At?  tends 
to  zero, 


_      _A    —((M\    Ar 
\dvj  Q 


.  (la), 


since  o-Av  is  a  small  quantity  of  the  second  order. 

Now  let  the  gas  warm  up  to  its  initial  temperature,  and  let  the 
recorded  pressure  be  j>2-     Then,  as  before, 

where  I -2)    refers  to  isothermal  change. 

/dp\ 

From  (la)  and  (2):   SL^LP?  _  V^: 
Ih  —  Po      (dp 

\dr/  e 
by  Reech's  theorem. 

This  equation  is  due  to  Moutier  (1880) ;  the  method  has  been 


(3), 


IDEAL   AND  PERMANENT  GASES  145 

used  by  Maneuvrier  and  Fournier  (1895-97),  and  by  Worthing 
(1911). 

Moutier's  equation  evidently  applies  to  any  gas,  whether  it 
obeys  the  laws  of  ideal  gases  or  not,  provided  Ar  is  small. 

(2)  Ideal  Gases.  —  The  state  of  unit  mass  of  an  ideal  gas,  under- 
going adiabatic  compression  or  expansion,  is  completely  defined 
by  the  equations 


K  — 


p 

In  the  deduction  of  these  it  has  been  assumed  that  : 
(i.)  The  gas  conforms  to  the  equation  pr  =  rT. 
(ii.)  The  specific  heat  at  constant  volume  is  constant  over  the 
range  of  temperature  T2  —  TI. 
By  taking  logarithms  we  find  : 

—  l»g  PI  _  tog  T2  —  log  TI    ,   ^ 
log  vi  —  log  vz      log  1-1  —  log  r2. 

log  p*  —  log  PJ 


PI  li 

The  experimental  realisation  of  adiabatic  conditions  is  difficult  ; 
heat  is  always  transferred  between  the  gas  and  its  surroundings 
by  conduction  and  radiation,  and  the  usual  plan  is  to  make  the 
changes  of  volume  occur  so  rapidly  that  the  heat  transfer  is 
negligibly  small. 

KxampJes.  —  (1)  A  gas  is  contained  in  a  vessel  furnished  with  a  stopcock, 
and  is  under  a  pressure  pi  slightly  greater  than  atmospheric  pressure  P. 
The  tap  is  opened,  so  that  the  gas  expands  rapidly  to  atmospheric  pressure, 
and  is  at  once  reclosed.  When  the  remaining  gas  has  attained  its  original 
temperature,  the  pressure  is  pz-  Show  that  : 


log  pi  —  logp.2 

(Clement  and  Desormes,  1819). 

(2)  "With  the  same  apparatus  as  in  example  (1),  the  absolute  temperature 
of  the  gas  in  its  initial  state  was  TI.  The  tap  was  then  opened  so  that  the 
gas  rapidly  expanded  to  atmospheric  pressure,  and  the  temperature,  deter- 
mined immediately  after  expansion  by  a  platinum  resistance  thermometer, 
or  a  thermo-element,  in  the  centre  of  the  vessel,  was  T-2.  Show  that  : 

K  _  log  Pi  —  tog? 


(Lummer  and  Pringsheim,  1898). 
T. 


146  THERMODYNAMICS 

76.     The  Velocity  of  Sound  in  Gases. 

A  conipressional  wave  is  a  disturbance  propagated  through  a 
medium  such  that  portions  of  the  latter  are  alternately  compressed 
and  expanded  in  the  path  of  the  disturbance.  If  the  wave  is  of 
such  a  kind  that  an  observer  who  moves  in  its  direction  of  pro- 
pagation with  the  same  velocity  as  the  wave  sees  no  change  in 
the  appearance  of  the  wave,  it  is  said  to  be  of  permanent  type. 

Newton  (1686)  first  calculated  the  velocity  of  propagation  of  a 
conipressional  wave  of  permanent  type  in  an  elastic  medium,  and 
arrived  at  the  general  formula  : 

w2  =  e/p          .         .         .         .       (1), 

where  e,  p  are  the  elasticity  and  density  of  the  medium  in  the 
unstrained  condition  (cf.  Lamb,  The  Dynamical  Theory  of  Sound, 
1910,  Chap.  VI.). 

The  passage  of  a  sound  wave  along  a  tube,  so  that  no  energy 
is  dissipated  by  friction,  is  an  example  of  a  conipressional  wave 
of  permanent  type,  and  Newton  applied  his  equation  (1)  to  deter- 
mine the  velocity  of  sound  in  air.  For  this  purpose  he  took  e  as 
the  isothermal  elasticity  of  air,  which  is  equivalent  to  assuming 
that  the  temperature  is  'the  same  in  all  parts  of  the  wave  as  that 
in  the  unstrained  medium.  Since  air  is  heated  by  compression 
and  cooled  by  expansion,  the  assumption  implies  that  these 
temperature  differences  are  automatically  annulled  by  conduction. 
Taking  the  isothermal  elasticity,  we  have : 

-'  ....     (la) 

We  may  call  un,  calculated  for  any  medium,  the  Newtonian 
relocity  of  sound. 

If  the  fluid  obeys  Boyle's  law,  a  case  approximately  realised  by 
atmospheric  air,  we  have 

efl  =  J>f 

•  '•  H»  =  Vp/p  =  A/pr. 

This  equation  gives  for  the  velocity  of  sound  in  air  at  0° 
280  metres  per  second  instead  of  331,  as  obtained  by  experiment. 
The  discrepancy  was  explained  by  Laplace  (1822),  who  pointed  out 
that  in  the  sound  wave  the  changes  of  volume  are  so  rapid  that 
the  conditions  are  adiabatic,  and  not  isothermal.  Hence  e  =  fQ, 


IDEAL   AND  PERMANENT   GASES  147 

Prom  (la)  and  (2)  we  obtain 


by  Reach's  theorem.    This  equation  is  true  for  any  fluid,  and  may 
find  application  in  the  future. 

If  the  fluid  is  a  gas  obeying  Boyle's  law  : 

*Q  =  KP 

.'.  «  =  V^        .        "...    (4) 

The  ratio  of  the  specific  heats  is  therefore  determined  by  (3) 
from  a  measurement  of  the  velocity  of  sound  in  the  gas  at  a 
particular  temperature,  provided  the  characteristic  equation  of 
the  fluid  is  known. 

In  the  deduction  of  Newton's  equation  (1),  it  is  assumed  that 
the  amplitude  of  the  vibration  is  small,  so  that  the  kinetic  energy 
of  the  moving  gas,  which  is  proportional  to  the  square  of  the 
amplitude,  is  negligible.  When  this  assumption  is  not  made,  the 
differential  equations  offer  formidable  difficulties  in  the  way  of 
solution.  The  problem  has  been  attacked  by  Riemann  and  by 
Earnshaw  ;  the  former  also  pointed  out  that  discontinuities  may 
exist  in  the  motion.  The  theory  has  also  been  studied  by 
Hugoniot,  whose  results  have  been  applied  to  the  analogous 
problem  of  the  propagation  of  explosion  waves  by  Jouguet 
(cf.  Riemann-Weber,  Partielle  D[fferentialgleichungen,  II.  ;  Jouguet, 
Journ.  d<>  Mathem.  1905,  p.  347  ;  1906,  p.  6  ;  Crussard,  Bull. 
Soc.de  Vindust.  mineral.  ,6,  1907;  Gyozo  Zeniplen,  P/»##.  Zeitschi; 
13,  498,  1912). 

77.     External   Work  in  Expansion. 

For  the  element  of  external  work  in  an  infinitesimal  expansion 
we  have  generally  : 

8A  =  pdi: 
(1)  Isopiestic  Expansion  :  p  =  const. 


r 

=  \  pdv  =  p(c2  — 


(1) 


i.e.,  Ap  is  represented  by  a  rectangular   area  on  the   indicator 

L  2 


148  THERMODYNAMICS 

diagram.     It  follows  from   the   characteristic   equation   that   T 
must  increase  along  with  v. 

(2)  Isothermal  Expansion :  T  =  const. 

J   n 


For  a  mol  :  A'    =  BT  /n 


If  QT  is  the  heat  absorbed,  we  see  from  §  73  (1),  that  : 

AT  =  QT, 
as  is  indicated  by  the  theorem  of  Joule  (§  71). 

AT  is   represented    on   the   indicator   diagram    by    the    area 
enclosed  by  the  v  axis,  the  ordinates  v  =  v\,  v  =  ?-2,  and  the 
rectangular  hyperbola. 
(8)  Adiabatic  Expansion  : 

SQ=0, 
.  *  .  dAQ  =  pdv. 

But  jn*  =  constant  =  k  [cv  const.]. 


:-.Aa=p<fe=tf| 

J    VI  J     Vl 


Also  :  Ay  =  — -j  (2)31-2  —  j>i«i)  =  —  -^-y  (T2  —  TO         •     (8«) 
-,  for  a  mol,  A' = -^  (T2  -  TO (36) 

K  X 

The  work  done  b}7  an  ideal  gas  of  constant  specific  heat  in 
p  passing  from  one  isotherm  to  another  is 

the  same  for  all  adiabatic  paths,  is 
independent  of  the  initial  or  final  pres- 
sures or  volumes,  and  is  proportional  to 
the  difference  of  temperature  between 
the  isotherms. 

If  we  put  r  =  c'j,  —  c,.,  and  Cj,/c,.  =•  K, 
in  (3a),  we  find  that  the  work  done  on 

o     be          c  ~d          "  adiabatic  expansion  is  cr(T2  — TO,  i.e.,  the 
FIG.  22.  whole   of    the    external    work    is    done 

at  the  expense  of  the  intrinsic  energy  of  the  gas. 


IDEAL   AND   PERMANENT   GASES  149 

Example.—  An    ideal    gas  of   constant    specific    heat  is  taken  round  a 
reversible  Carnot's  cycle,  represented  by  four  curves  (Fig.  22): 
AB  an  adiabatic  pv«  —  clf  .-.  pava*  =  c\, 
BO  an  isotherm    pv  =  c2,  .*.  pcve    =  c.2, 
CD  an  adiabatic  pv*  —  cs,  .-.  pevcK  =  c3, 
DA  an  isotherm  /w  =  ct,  .  •.  paVa     —  c±. 
Find  the  work  done  in  the  cycle. 

Work  done  —  area  of  cycle 
=  ABCD 

=  -  AaiB  +  ECcb  +  ODrfc  -  DrZaA. 
But  Aa&B  =  CVdc, 

since  the  work  done  in  passing  from  one  isotherm  to  another  is  the  same 
along  all  adiabatic  paths, 

.-.  work  =  ECcb  -  DdaA. 


But  pbvb  =  peve  =  ca  ;  pava  K  =  pbvbK  =  Cl, 

•••   Vf   -1  =Ci/Cg. 

Similarly  vc  K  ~  l  =  c3/c2,  .  •.  (ob/ve]  K  ~  1  =  Cl/cs. 

Similarly  (va/vd}  "~l  =  ci/cs. 


. 

Vb  K   —  1          Ci 

cjnvjL  =  ^^lnCJ, 

Va          K   —    1          Cl 

.  •.  work  done  per  cycle  =  —  ^-~  In  -. 

78.     Entropy   of  an   Ideal   Gas. 

Let  unit  mass  of  an  ideal  gas  pass  reversibly  from  the  state 
r,  T  to  the  state  v  +  dr,  T  +  <?T. 
From  the  general  equation  : 

(l,i  =  TV*  —  jM  * 

,1,,  _|_  pfr   I     .         .         .         .     (1) 
we  have  <fx  =  --  ^  —    j 

Now  -  flu-  —  c//T     '     .         .         .         .     (2) 

and  p  =  —  -•    .         .         .         .         .     (3) 


For  a  finite  change  of  r  and  T  : 

+  rlnv  -f  const. 


=     ^JJT- 


1.10  THERMODYNAMICS 

For  a  mol  of  gas  : 

S  =  I  ^-  +  Elm-  +  Const.  (6) 


-JSf 


Corollary.  —  If  the  volume  of  a  mol  of  an  ideal  gas  is  increased 
tenfold    at    constant    temperature    the    entropy    increases    by 

4*57  ?:  C.a  ',  independently  of  the  nature  of  the  gas. 

Equations   (5),  (6)  define  without   ambiguity   the    change  of 
entropy  in  any  specified  change  of  state. 


+  *  ...      .     (7) 

Tl 

We  first  assume  that  the  specific  heat  at  constant  volume  is 
independent  of  temperature  (Clausius). 

Then  s  =  crlnT  +  rlnv  +  const.         .         .         .    (8) 

If  we  put  T  =  1,  r  =  1,         .         .         .         .    (9) 

const.  =  [.S-]T  =  i  =  -so,  say, 

where  s0  is  the  entropy  in  the  standard  state  : 

.-.  «  =  80  +  c>T  +  rlnv  =  80  +/wTW  .         .     (10) 
Similarly  : 

S  =  S0  +  C>T  +  ~Rlnv         .        .        .    (11) 
where  S0  =  M*0,  C,  =  Mcr,  R  =  Mr    .        .         .    (12) 

M  being  the  molecular  weight. 

If  we  eliminate  v  or  T  from  (10)  by  means  of  the  equation  : 

pv  =  rT 

we  find  expressions  for  the  entropy  in  terms  of  (p,  T)  or  (p,  v)  : 
8  =  80  +  cJnT  +  rlnr  =  *0  +  /wTW  \ 

=  *o'  +  (fr  +  /'X»T  -  rbip  =  so'  +  /wTf"  +  rp  ~  rl   .     (13) 
=  so"  +  (ce  +  r)lnr  +  c,hp  =  s0"  +  /?jTf"  +  ')/»  ) 
where  s0'  =  SQ  -\-  rlnr,  s0"  =  s0  —  cjur. 

If   we   multiply    the    equations   (13)   by   M   we    obtain    the 
expressions  for  a  mol.: 
S  =  So  +  Cr/nT  +  R/nr  =  S0  +  /»TcVrR 
=  So'  +  (C,  +  R)/»T  -  Rlnp  =  So'  +  /"T11.  +  Kp  ~  K        .     (14) 
=  So"  +  (Cr  +  R)/wr  +  CJnp  =  S0"  +  lnrf'*  +  Kp1'"       J 

where  So'  =  S0  +  R/«  ^  ;  S0"  =  S0  -  C,/»  ^  ; 

?'  =  specific  volume. 


IDEAL   AND   PERMANENT   GASES  151 

It  we  put  I/Mr  =  £,  the  volumetric  molecular  concentration, 
or  the  number  of  mols  per  unit  volume, 


.-.  S  =  So"'  +  CJnT  -  R/H£  =  So'"  +  /HTC,£  -  R     .     (15) 
where  S0'"  =  S0  -  R/nM. 

Example.—  If  a  mol.  of  an  ideal  gas  changes  reversibly  from  a  state  of 
10  litres  at  15°  C.  to  100  litres  at  50°  C,  show  that  the  increase  of  entropy  is 

4.949    S-cal-     if  Cv  =  3  cal. 


If  the  true  specific  heat  of  an  ideal  gas  is  a   function  of 
temperature  of  the  form : 

cv  =  a  +  26T 
it  can  easily  be  shown  that  its  specific  entropy  is : 

s  =  s0  4-  «7»T  4-  26  (T  —  1)  4-  rlnv 
and  the  entropy  per  mol  (M)  is  : 

S  =  S0  4-  alnT  +  2  £(T  —  1)  4-  Elm; 
where  a  =  Ma,  0  =  M6,  R  =  Mr. 

79.     Free  Energy  and  Potential  of  an  Ideal  Gas. 

By  definition  we  have,  for  any  system, 

the  free  energy  *  =  U  —  TS         .         .         .         .     (1) 

the  potential  4>  =  *  +  pV  =  U  —  TS  4-  p\   .     (2) 

For  unit  mass  of  an  ideal  gas : 

M  =  |c/?T  .         .         .         .         .     (3) 

n~ +'•'«*'  '  •  (4) 


-  T  -  rT/»»-  =  -  rflnr  +  (/'(T)     .     (5) 

=  -  rT  (Inr  -  1)  +  //'  (T)       .     (6) 
where  r/'(T)  =  »0  -  T-s0  +  Je(WT  -  TJ^  .         .         .         .     (7) 

is  a  function  of  temperature,  uQ,  s0  being  the  two  arbitrary 
constants  depending  on  the  choice  of  the  initial  states  of  energy 
and  entropy. 

If  the  specific  heat  cv  is  constant  we  find : 
.  =  « 

' 


152  THERMODYNAMICS 


,  r 

=  (MO  -  «o'T)  +  T(cv-lnT*  +  'jr')     .         .     (9), 
=  (MO  -  *o"T)  +  T(cv  -  InV"  +  ^J 
where  «o'  =  *o  —  rlnr;  s"  =  «0  —  c^wr. 

Similarly,  for  the  potential  : 

</>  =  MO  -  s0T  +  T(<v  -  /nT-rrO      v. 
=  ,,o-VT+T(c;)-/»T^-0         .         .     (10), 
=  MO  -  *0"T  +  T(cp 


where  cp  =  c,,  +  r  =  cv  + 

For  a  mol  of  gas  we  have  : 

M^  =  U0  -  S0T  +  T(C.  -  toT°n»B)  } 

=  U«-SoT-f  T(C.-JwT°^*p-B)         ;        .     (9a) 

=  U0  -  S0"T  +  T(CV  -  IniP*  +  V)    J 
and     M</>  =  U0  -  S0T  +  T(CP  -  /».T%B)  ^ 

=  U0  -  So'T  +  T(C,,  -  lnT°pp  ~  R)  .  (lOo)  , 

=  U0  -  S0"T  +  T(CJ(  -  Inv  Vc")         ) 
where  v  is  still  the  specific  volume. 

In  the  case  of  varying  specific  heat,  if  we  put  : 

cr  =  a  +  26T      ......    (11), 

then  u  =  MO  +  aT  +  6T2 

s  =  s0  +  aZnT  +  2?>  (T  -  1)  +  rZnv 

.  M/r  =  jt  —  Ta  =  («0  —  Ts0)  —  «T7nT  —  iT2  —  rTZra;  +  (a  +  27>)T  } 
<j)  =  ^-\-rT  =  (MO  -  Ts0)  -  aTZnT  -  />T2  -  rT(Z72  v  -  1)  +        [(12) 

(a  +  2fe)Tj 
For  a  mol  of  gas  : 
M*  =  (U0  -  TSo)  -  aTZnT  -  /3T2  -  ET/w  r  +  (o  +  2j8)T     1 

^1 


M^>  =  (U0  -  TS0)  —  oTZnT  —  /3T2  —  ET(/n«;  — 

Examples.  —  (1)  If  a  mol  of  gas  expands  isothermally  and  reversihly  from 
volume  Y!  to  volume  V2  tae  diminution  of  free  energy  is  : 

*&••-»—  t{*<sr-sffir}-,**% 

which  is  equal  to  the  maximum  work  (§  58). 

(2)  If  a  gas  obeys  Boyle's  law  the  diminution  of  potential  on  isothermal 
reversible  expansion  is  equal  to  the  diminution  of  free  energy,  and  both  are 
equal  to  the  maximum  work. 

(3)  The  heat  absorbed  in  the  isothermal  expansion  of  an  ideal  gas  is  equal 
to  the  increase  of  the  W  function. 

80.     Deviation  of  Gases  from  Boyle's  Law. 

Although  Boyle  himself  appears  to  have  thought  that  his  law 
did  not  hold  good  under  higher  pressures,  Despretz  (1827)  was 


IDEAL   AND  PERMANENT  GASES  153 

the  first  to  show  conclusively  but  qualitatively  that  all  gases 
were  not  equally  compressible.  Such  gases  as  carbon  dioxide 
and  ammonia  were  more  compressible  than  air,  whilst  above 
15  atm.  hydrogen  was  less  compressible.  Other  experimenters 
(Pouillet,  Dulong,  and  Arago)  obtained  conflicting  results,  and 
the  matter  was  left  in  uncertainty  until  Eegnault  (1847)  carried 
out  an  investigation  on  the  compressibility  of  gases  with  his 
customary  thoroughness  and  accuracy.  He  employed  pressures 

Pv  pv 


p  1000  2000  3000  atm 

FIG.  23. 

up  to  30  atm.,  and  found  that  no  single  gas  conformed  strictly  to 
the  law  of  Boyle.  With  the  exception  of  hydrogen  they  were  all 
more  compressible  than  they  ought  to  be;  hydrogen,  however, 
diminished  in  volume  less  rapidly  than  the  pressure  increased, 
and  was  ironically  styled  by  Regnault  "  un  gaz  plus  que  parfait." 
If  TO,  v  are  the  volumes  under  the  pressures  po,  p(p>  Po),  then, 
if  Boyle's  law  were  strictly  obeyed,  we  should  have  iwo/pv  =  1. 
Regnault  found,  however,  that  /Wo/pr  >  1  for  all  gases  except 
hydrogen,  for  which  p&ol'pv  <  1. 

Natterer  (1852)  made  a  series  of  unsuccessful  attempts  to 
liquefy  the  "  permanent  gases  "  by  exposing  them  to  enormous 
pressures  (3,000  atm.),  in  the  course  of  which  the  very  important 


154  THERMODYNAMICS 

fact  came  to  light  that  at  rri'i/  hitjlt  ;>/v.s-.s»/r.s  all  //a.sv.s  hi-han-  HI,/' 
Jiydt'oqcn  in  being  less  compressible  than  an  ideal  gasl  It  follows 
that,  at  some  intermediate  pressure,  all  gases  except  hydrogen 
will  obey  Boyle's  law  strictly ;  below  this  pressure  they  are 
more,  above  it  less,  compressible  than  an  ideal  gas.  If,  therefore, 
values  of  pv  are  plotted  against  p,  all  the  gases  investigated  by 
Natterer,  except  hydrogen,  will  give  curves  showing  minima.  It 
is  possible  that  the  hydrogen  curve  may  have  a  minimum  at  a 
very  low  pressure. 

These  results  were  confirmed  and  amplified  in  an  extensive 


pv 


P  1000  2000  5000 Atm. 

FIG.  24. 

series  of  researches  of  Amagat  (Ann.  Clrim.  Phys.  [5],  19,  435, 
1880,  [4],  28,  274,  1873,  29,  246,  1873;  [5],  22,  353,  1880;  28, 
480,  1880)  with  the  gases  02,  N2,  air,  H2,  C02,  C2H4,  CH4.  He 
found  curves  exhibiting  minima  with  all  gases  except  hydrogen, 
the  minima  shifting  to  the  right  with  rise  of  temperature. 
Amagat's  diagrams  for  hydrogen,  nitrogen,  and  carbon  dioxide 
are  shown  in  Figs.  23 — 25. 

The  investigation  of  gases  at  very  low  pressures  is  a  matter  of 
considerable  difficulty,  and  is  exposed  to  various  sources  of  error 
(cf,  Travers,  Experimental  Study  of  Gases). 


IDEAL   AND  PERMANENT   GASES 


155 


1-0 


It  was  therefore  not  surprising  that  very  contradictory  results 
were  obtained.     Thus  Mendeleeff  and  Kiripitscheff  thought  that 
pr  steadily  decreased 
with     the     pressure, 
whilst  Amagat  found 
that  air  obeyed  Boyle' s  2-0 
law  almost  exactly  at 
very  low  pressures. 

The  most  accurate 
experiments  in  this 
region  are  probably 
those  of  Lord  Ray- 
leigh  (1901-2),  who 
examined  hydrogen, 
nitrogen,  and  oxygen 
at  pressures  of  O'Ol 
mm.  to  1*5  mm.  in  a 
very  ingenious  appar- 
atus. He  found  that 
if  any  deviation  from 
Boyle's  law  existed  at 
these  low  pressures  it 
was  within  the  limit 
of  experimental  error,  remarking  that  "  experimental  errors 
could  not  well  transform  an  apparently  complex  to  a  simple 
relationship." 

The  condition  for  the  strict  validity  of  Boyle's  law  is  obviously 

' . . .'  (¥),=<>  •   •  •  • (1) 

The  following  results  have  been  obtained  relating  to  the  altera- 
tion of  pv  with  pressure : 

(1)  For   different   gases   at  0°  and  with  the  same  range  of 

pressures,     j-  -  ranges  from  a  relatively  large  negative  value  for 

the  easily  liquefiable  gases  to  a  small  positive  value  for  hydrogen. 

(2)  For  the  same  gas  at  different  temperatures  and  the  same 

range  of  pressures,  (  y   ^  changes  from  a  negative  value  at  low 
temperatures  to  a  small  positive  value  with  rise  of  temperature. 


500 

Pressure  in  Aim. 
FIG.  25. 


4000 


156 


d(pv) 


THERMODYNAMICS 

0°  52°  100° 

—  -000571  ±0  +  -000347. 


(8)  For  the  "  permanent "  gases  at  constant  temperature 

«z* 

remains  nearly  constant  from  very  low  pressures  up  to  pressures 
of  8—4  atm. 

The  last  conclusion  was  drawn  by  D.  Berthelot  from  the  results 
of    Rayleigh    and    Leduc,    which    showed   that   at  very   small 


PV 


450  SSO 

FIG.  26. 


pressures  (1*5 — O'Ol   mm.)  the  value  of   p\r\lp-n'z  is    constant, 
and  apparently  equal  to  1  within  the  limits  of  experimental  error. 


It   is   probable,   however,   that  they  really  prove  that 
small  and  constant  at  low  pressures : 

=  const.  =  a  . 


dp 


(1) 
(2), 


dp 

.'.  pi-  =  ap  -f-  b 
where  b  is  a  constant. 

pi-  is  thus  a  linear  function  of  p,  and  the  curves  in  which  p,  jn- 
are  abscissa  and  ordinate  are  straight  lines  inclined  at  angles 


IDEAL   AND  PERMANENT  GASES  157 

tan~1(a)  to  the  p  axis.     If  the  gas  obeys    Boyle's   law,  a  =  0, 
.-.  pv  =  b,  and  the  lines  are  parallel  to  the  p  axis. 

The   product  pv   tends   to   a   finite   limit    as    p    is    reduced 
indefinitely  : 

Lim  (pv)  =  pQv0  =  b  .         .     (3), 


which  is  the  ordinate  of  intersection  with  the  pv  axis. 

The  extrapolation  to  zero  pressure  may  be  effected  if  two 
measurements  of  pi  at  small  pressures  have  been  made,  provided 
the  gas  satisfies  equation  (1).  If  this  is  not  the  case  {e.g., 
hydrogen  chloride,  carbon  dioxide),  a  number  of  points 
must  be  fixed  on  the  curve,  especially  in  the  low  pressure 
region. 

The  curves  for  neon,  helium,  and  oxygen  are  shown  in  Fig.  26. 
At  0°  C.  neon  gives  a  straight  line  sloping  downwards  to  the  pc 
axis;  it  is  a  "gaz  plus  que  parfait."  Helium  gives  a  perfectly 
horizontal  line,  although  this  strict  conformity  to  Boyle's  law 
may  hold  good  only  at  0°  C.  ;  oxygen  gives  a  straight  line  sloping 
upwards  to  the_pr  axis.  Carbon  dioxide,  hydrogen  chloride,  and 
the  more  coercible  gases,  give  lines  showing  distinct  curvature, 
more  marked  in  the  lower  pressure  region. 

The  definition  of  molecular  weight  given  in  §  68  refers  only  to 
ideal  gases  ;  in  the  case  of  gases  which  do  not  follow  the  gas  laws 
it  is  obvious  that  Avogadro's  theorem  is  no  longer  strictly  applic- 
able. For  if  we  suppose  that  equal  volumes  of  two  gases  contain 
a  molecular  weight  of  each  under  specified  conditions  of  tempera- 
ture  and  pressure,  these  volumes  will  not  remain  exactly  equal 
if  the  temperature  and  pressure  are  altered,  for  each  gas  exhibits 
its  own  peculiar  deviations  from  the  gas  laws,  and,  since 
equiuiolecular  weights  are  now  contained  in  different  volumes 
under  like  conditions  of  temperature  and  pressure,  it  is  evident 
that  the  theorem  of  Avogadro  is  no  longer  valid.  Since  the 
deviations  are  only  small,  a  determination  of  the  normal  density 
gives  a  very  approximate  value  of  the  molecular  weight,  and  this 
method  has  long  been  in  use  to  decide  between  possible  multiples 
of  numbers  obtained  by  exact  gravimetric  methods.  In  recent 
years,  however,  a  method  of  finding  the  molecular  weight  of  a 
gas  directly  from  the  density  with  an  accuracy  rivalling  that 
of  the  gravimetric  methods  has  been  elaborated,  and  will  next 
be  considered. 


158  THERMODYNAMICS 

The  compressibility  coefficient   is   the   deviation  from   Boyle's 
law  per  unit  pressure  : 


at  the  pressure  p. 

If  (1)  applies,  the  coefficient  between  the  limits  1  and  0  atm.  is  : 


.         . 
—  PO) 

The  variation  from  Avogadro's  theorem  may  be  expressed  in 
terms  of  the  density  per  unit  pressure.  Let  m  be  the  weight  in 
grams  of  v  litres  of  a  gas  under  a  pressure  p  atm.  at  0°  C.  The 
density  is  w/r  and  the  density  per  unit  pressure  mjpv.  When 
/>  =  1  the  expression  represents  the  absolute  normal  density, 
and,  when  p  is  very  small,  the  absolute  limiting  density.  At 
intermediate  pressures  the  densities  per  unit  pressure  are  : 
in  in  m 

P&i  PsF*  p&a 

With  diminishing  pressure  these  values  remain  constant, 
decrease,  or  increase,  respectively,  according  as  the  gas  obeys 
Boyle's  law  (He),  or  is  more  (Ne)  or  less  (02)  compressible  than 
this  requires. 

The  ratio  of  the  absolute  limiting  densities  of  two  gases  is  : 


where  mi,  ?»2  are  any  masses,  and  Owo)i,  Ow0)a  tne  limiting 
values  of  pv  for  these. 

Since,  however,  the  ratio  of  the  absolute  limiting  densities  is 
the  ratio  of  the  masses  of  equal  volumes  of  the  two  gases  when 
the  latter  are  in  the  ideal  limiting  state,  it  follows  from  Avogadro's 
theorem  that  this  is  also  equal  to  the  ratio  of  the  molecular  weights  : 
MI  _  >ni 


,~ 
M2  ~~  wj   '        - 


.But  if  DI  is  the  normal  absolute  density  (at  N.T  P.)  : 


Pi 
where  pi  =  1  atm.,  and  similarly  for  D2. 


D!  X 

.    Mi  _  \poroj  i  ,o, 

•ff'~D,T(M 

\poVoJt 


IDEAL   AND   PERMANENT   GASES  159 

If  (1)  applies,  the  equation  may  be  written  : 

Mj 
M2 


or  M!  :  M2  :  M3  :  .  .  . 
for  any  number  of  gases.  Berthelot  calls  this:  "1'echelle  des 
poids  moleculaires."  If  one  molecular  weight  is  fixed  by  arbitrary 
choice,  the  rest  are  determined,  and  02  =  32  is  taken  as  standard. 
Then  DI  X  32  is  the  normal  density,  and  the  limiting  density  or 
molecular  weight  is  : 


,  ,       -^ 
normal  density  X 


The    products    Di(l  —  aj),   etc.,   are    the    limiting    absolute 
densities,  32  Di(l  —  al0\,  etc.,  the  limiting  densities. 
Also: 

M;  M2  M3 


the  molecular  volume  of  an  ideal  gas  at  N.T.P. 

The  molecular  weight  of  a  gas  may  therefore  be  determined, 
with  respect  to  02  =  32  as  standard,  from  the  data : 
(a)  Normal  density  of  the  gas, 
(6)  Compressibility  of  the  gas, 
(c)  Compressibility  of  oxygen. 

Berthelot  showed  that  the  mean  compressibility  between  1  and 
2  atm.  does  not  differ  appreciably  from  that  between  0  and  1  atm. 
in  the  case  of  permanent  gases,  and  either  may  be  used  within 
the  limits  of  experimental  error.  But  in  the  case  of  easily 
liquefiable  gases  the  two  coefficients  are  different.  According  to 
Berthelot  and  Guye  the  value  of  aj  can  be  determined  from  that 
of  aj  by  means  of  a  small  additive  correction  derived  from  the 
critical  data,  and  the  linear  extrapolation  then  applied;  Gray 
and  Burt  consider,  however,  that  this  method  may  lead  to 
inaccuracies,  and  consider  that  "  the  true  form  of  the  isothermal 
can  only  be  satisfactorily  ascertained  by  the  experimental  deter- 
mination of  a  large  number  of  points,"  followed  by  graphical 
extrapolation. 


160  THERMODYNAMICS 

Examples.—  (I)  Molecular  weight  of  hydrogen  (linear  relation)  : 

mol.  \vt.  =  normal  density  X  — 

(1  -  «j)o2 


=  2-01413. 
(2)  Molecular  weight  of  hydrogen  chloride  (non-linear  relation)  : 

02 


do2           (Pivi\ 

(Po*>o)o,(extrap,) 

(<42)o  =  do2  X  ( 

1-42900     139,628 

139,769 

1-42756 

138,959 

139,087 

1-42768 

56,526 

56,311 

1-42760 

mean     1-42762 

HC1   (/HOI   (JMI''I)HCI    (P»*O)HCI    (^HCI)O  =  ^HCI  X  ( 
1-63915   54,803    55,213  1-62698 


.-.  Mol.  wt.  HOI  =  32  x  ^HCUO  =  32  ^1^™  i=  56.469 


The  molecular  weights  calculated  in  this  way  agree  very  closely 
with  the  so-called  "chemical  molecular  weights,"  derived  from 
chemical  analysis,  and  the  method  therefore  rivals  the  latter  in 
accuracy,  without  the  attendant  complications  attaching  to 
chemical  operations  with  pure  substances. 

(Lord  Rayleigh,  Phil.  Trans.  198,  417,  1902;  Proc.  Roy.  Hoc.. 
73,  153,  1904.     D.  Berthelot,  Journ.  de  phys.  [3],  8,  263,'  1899; 
Sitr  les  thermometres  a  gaz,  Trav.  et  Mem.  dn  Bureau  Int.  Poids  ct 
Mesures,  1907,  vol.  13  ;  Zeitschr.  Elektrochcm.  34,  621,  1904.) 
81.     Deviations  from  the  Law  of  Dalton  and  Gay-Lussac. 

The  fact  that  the  coefficients 

1  dv 


differ  from  each  other  for  the  same  gas,  and  among  themselves 
with  different  gases,  was  ascertained  by  Magnus  (1842)  and 
Regnault  (1842). 

With  increasing  density,  /3  increases  to  a  maximum  and  then 
decreases.  With  increasing  temperature,  and  the  same  initial 
density,  it  decreases. 

With  increasing  pressure  a  increases  to  a  maximum,  and  then 
decreases.  It  also  increases  to  a  maximum  with  increasing 
temperature,  and  then  decreases. 


IDEAL  AND   PERMANENT  GASES 


161 


It  has  often  been  supposed  that  at  very  high  temperatures  all 
gases  would  behave  normally,  i.e.,  would  approach  a  limiting  ideal 
state.  As  a  matter  of  fact  the  deviations  appear  to  be  influenced 
by  the  density  of  the  gas,  and  disappear  at  infinitely  small 
densities  whatever  the  temperature  may  be.  Thus  a  saturated 
vapour  at  very  low  temperatures  may  behave  like  a  permanent 
gas,  on  account  of  its  very  small  density. 

Regnault  expressed  the  opinion  that  the  coefficients  of  expan- 
sion of  all  gases  would  probably 
approach  the  same  limiting  value 
as  the  pressure  was  diminished. 

If  pv  is  a  linear  function  of  p, 
a  and  /3  approach  a  common  limit, 
7,  as  p  tends  to  the  value  zero  : 
Lim  a  =  Lim  ft  =  7. 
p  -»  0       p  ->  0 

Let  Aa,  B6  (Fig.  27)  be  the 
pv  lines  corresponding  to  tem- 
peratures 0i,  6-2  for  a  molecular 
weight  of  the  gas. 

Taking  any  pressure  p',  draw  O'A'B'  at  right  angles  to  the 
j>-axis,  join  OA'  and  produce  to  meet  B6  in  B".  Draw  B"0"  at 
right  angles  to  the  j?-axis. 

B&,  Aa  are  lines  of  constant  temperature  (isotherms), 

O'B,  0"B"  are  lines  of  constant  pressure  (isopiestics), 

OA'B"  is  a  line  of  constant  volume  (isochore), 


because 


tan  O'OB"  =  ^  =  r  =  const. 
P 

1  r"-t;'  p't"-pc 

a 7f    •  ~, (P  COnst.)  =  — 

02  —  01          v'      u  /IT' 


O'B'  -  O'A'      A'B' 


O'A' 
1 


O'A' 


00"  -  00' 
OO' 


1         OB"  -  OA' 
~  02  -  0r        OA' 


A'B" 


As  p   diminishes  indefinitely,  corresponding  to   rotation  of 
T.  M 


162  THERMODYNAMICS 

OA'B"  to  coincidence  with  OAB,  a  and  /3  tend  to  the  common 
limit 

1         AB 
V  -  02  -  0,  •  OA' 

and  since,  by  definition  of  molecular  weight,  A  and  B  are  the 
same  points  for  all  gases,  7  is  independent  of  the  nature  of  the 
gas,  and  is  the  constant  for  an  ideal  gas.  Hence 

1 

This  provides  a  method  of  determining  absolute  temperatures, 
for  we  have  proved  in  §73  that 

1 

a 

where  a  is  the  coefficient  of  expansion  of  an  ideal  gas,  i.e.,  the 
limiting  value  7  determined  in  the  manner  just  described. 
In  this  way  Berthelot  found 

1/7  =  273-09 
.'.  T  =  6  +  273-09, 

The  method  in  use  previous  to  Berthelot's  depended  on  the 
results  of  the  "  Porus  Plug  "  experiments  of  Joule  and  Kelvin. 
(Kelvin's  Math,  and  Phys.  Papers,  Vol.  I.) 

82.     The  Joule-Kelvin  Experiment. 

The  experiments  of  Dr.  Joule  and  Lord  Kelvin,  instituted 
with  the  object  of  testing  the  validity  of  Mayer's  hypothesis 
(dU/dv  =  0),  were  carried  out  in  the  former's  brewery  at  Man- 
chester during  the  years  1852-62.  The  principle  of  the  method 
is  very  simple,  although  great  difficulties  were  encountered 
in  its  realisation.  A  stream  of  gas,  under  a  constant  pressure 
higher  than  atmospheric,  was  forced  continuously  through  a 
porous  plug  of  cotton-wool,  or  silk,  supported  in  a  boxwood  tube. 
The  temperatures  of  the  gas  before  and  after  passing  the  plug  were 
determined  and  after  various  corrections  the  change  of  tempera- 
ture experienced  by  the  gas  was  found.  By  the  friction  of  the 
air  in  passing  through  the  plug  heat  is  produced,  and  at  the 
same  time  heat  is  absorbed  by  the  expansion.  The  difference 
will  be  the  heat  evolved  or  absorbed.  The  heat  produced  by 
friction  is  equal  to  the  work  which  could  have  been  obtained 
if  the  expansion  had  occurred  without  friction. 


IDEAL   AND  PERMANENT   GASES 


163 


Let  _pA,  rA,  TA  and  pB,  VB,  TB  be  the  pressures,  specific  volumes, 
and  absolute  temperatures  of  the  gas  before  and  after  passing  the 
plug  respectively  (Fig.  28). 

We  may  suppose  the  transition,  as  actually  effected  in  the 
experiment,  brought  about  by  pass- 
ing a  volume  rA  of  gas  through 
the  plug  by  means  of  a  piston 
exposed  to  the  constant  pressure  p^ 
and  then  allowing  it  to  push  out, 
through  a  volume  VB,  the  piston  on 
the  other  side  exposed  to  the  constant  pressure  J)B. 

If  Q  is  the  heat  absorbed  in  the  vicinity  of  the  plug,  «A,  UB,  the 
internal  energies  of  unit  mass  of  gas  in  the  states  A  and  13, 
referred  to  some  standard  state  as  zero,  then 

Q  =  ("A  —  MB)  +  (We  —p&A- 
Now  put  pA  =  pB  +  bpB, 

VA  =  VB  +  SrB, 
TA  =  TB  +  8TB, 

and  drop  the  suffixes  : 

8Q  =  bu  +  b(pv)  =  b(u  +  in-)  =  bw          .         .  (1) 
If  there  is  no  heat  exchange  on  passing  the  plug,  i.e.,  the 
process  is  adiabatic,  as  Joule  and  Kelvin  assumed, 

SQ=0           ....  (2), 

.-.  bit,  =  —  6(>-)   ....  (3), 

or        b(u  +  2>tf  =bw  =0         .         .         .     (3«). 


But 


(4), 


/.  Cj,ST  +  lpbp  +  rbp  =  0 


or 


Now 


,    . 
and  since 


d-v 


dv 


.  (5). 

.  (6), 

.  (7), 

.  (8), 


M  2 


]  64  THERMODYNAMICS 

we  have 

^TaCPfj-fOfc 

or 


from  which 

(P)  .i.(i*_.)    .    .     .  (9.), 

\dp  I  „.       ey)  V     91 

giving  the  heating  or  cooling  effect  per  atmosphere  difference  of 
pressure  on  the  two  sides  of  the  plug. 

[If  the  gas  is  an  ideal  yas,  pv  —  ?'T     .*.  T  ^  -t  =  *-  T  =  v 


--      =  0  from  (9a).] 
.op 


If  instead  of  (4)  we  write  : 


then,  from  (3  a)  : 


an  equation  due  to  Jochmann  (1859). 

We  can  also  write  the  Joule-Kelvin  equation  (9a)  in  the  form 


(dT),,;  is  the  rise  in  temperature  on  passing  the  plug,  which 
may  be  measured  on  a  gas  thermometer,  because  the  absolute 
degree  has  been  denned  as  equal  to  the  Centigrade  degree. 

We  shall  call  (dT)w  =  x,  the  plug  effect,  and  hg-J    the  J<>  nlc- 

Kelvin  effect.     It  was  found  that  in  the  case  of  oxygen,  nitrogen, 
air,  and  carbon  dioxide,  there  was  a  cooling  effect,  so  that  x  is 


IDEAL  AND   PEEMANENT   GASES  165 

negative  for  these  gases,  whilst  in  the  case  of  hydrogen  there  was 
a  very  slight  warming  effect,  so  that  x  is  positive  for  this  gas. 
Further,  x  was,  at  a  constant  temperature  of  the  entering  gas, 
proportional  to  the  fall  of  pressure  through  the  plug  : 

—  *  —  =  (^\   =  const.  =  o>°  C.  per  atm.        ,     (10) 
PA.—I*      \dp/» 
Equation  (9a)  can  be  written  : 

T^  =  oJI»  +  r     ....     (11) 
and  (91)  is  : 


(11)  and  (12)  are  differential  equations  for  T  in  terms  of  the 
magnitudes  p,  r,  cv,  cp,  co,  x,  but  as  these  cannot  be  directly 
determined  as  functions  of  T,  the  equations  cannot  be  integrated. 
We  may,  however,  suppose  T  to  be  some  function  of  the  Centigrade 
temperature  0,  measured  on  a  thermometer  containing  the  gas 
under.  investigation,  and  put  : 

dp       dp    W       ,  dv       dv     dO  /1Q\ 

gf  =  3§-sTand3T  =  5raT      •      • 

52  or  ^  is  now  measurable  by  separate  experiments  on  the  con- 
oa       off 

stant  volume,  or  constant  pressure,  thermometer  with  the  gas, 
and  we  shall  have 


w 

If  /dT 


cr,  Cp,  ^,  ^,  p,  pv,  and  x  are  now  all  expressible  in  terms  of  6, 

and  the  equations  are  integrable,  giving  T  as  a  function  of  6 
(Jochmann). 

Joule  and  Kelvin  proceeded  differently.  They  found  that  u  increased 
(i.e.,  the  cooling  diminished)  with  rise  of  temperature  of  the  entering  gas,  and 
they  assumed  that  in  the  equation  representing  a>  as  a  function  of  temperature, 
the  absolute  temperature  could  be  replaced  by  the  approximate  value 
(273-7  +  0).  In  this  case  the  experimental  results  agreed  with  the  formula  : 


(273-7  +•)« 
where  a  is  a  constant  for  a  particular  gas. 


166  THERMODYNAMICS 

With  a  mixture  of  gases,  the  cooling  effect  was  less  than  the  average 
effect  for  the  pure  gases. 

Equation  (14)  with  (10)  is  now  integrable. 

The  results  agree,  however,  somewhat  better  with  an  equation  proposed 
by  Kankine  (1854) : 


In  (16),  A/Vp  is  taken  as  constant  =  a. 

Let  us  now  suppose  that 

T  =  AS  .         .....         .     (18) 

where  3  =  0  +  - 

is  the  absolute  gas  temperature,  for  the  particular  gas,  which 
corresponds  with  T°  abs. ;  a  is  the  coefficient  of  expansion. 

Then  T(^)=V 

in  which  the  plug  effect  is  (y-  j     as  actually  measured  with  the 
gas  thermometer. 
But  cp  \  — 

where  cj  is  the  specific  heat  of  the  gas  at  constant  pressure  as 
measured  by  a  thermometer  filled  with  the  same  gas ;  hence  : 

?T_ «H = *L_     _  =  ^(1_</^    ' 

rli.<^\    I      M         »W 


since   "J«(gl)     i8    small    compared    with    unity    (cf.   H.   M., 

Appendix) ;  hence,  on  integration  : 

/• 

ZnT  =  Inv  -  \CJL  ^  dv  +  C   .         .         .     (21) 
J   v  dp 

/*V2 

The  value  of       £2'  .  ^-dv=  p,  say,  can  be  calculated  if  the 
J'v«       ^ 

value  of  ^J     is  known  from  the  experimental  results,  and  in 
this  case : 


IDEAL  AND  PERMANENT   GASES  167 

where  e~M  is  the  factor  which  reduces  a  ratio  of  two  directly 
measured  gas  temperatures  to  the  ratio  of  the  absolute  therrno- 
dynamic  temperatures.  (Cf.  Callendar,  Phil.  Mag.  [vi.],5,  48, 
1903.  E.  Buckingham  adopts  a  different  treatment,  assuming 
that  the  process  is  isothermal  rather  than  adiabatic  :  cf.  Phil. 
Mag.  [vi.l,  6,  518,  1903.) 

The  Joule-Kelvin    experiment    has    recently  (1909)   been   repeated  by 
J.  P.  Dalton,  who  finds,  for  the  plug  effect  in  air  : 

(<iT)r  =  0-273  (/>-!)-  0-000208  (/,*  -  1), 

where  p  =  initial  pressure  in  atm.  The  initial  temperature  was  O3  C.  The 
results  were  in  agreement  with  a  characteristic  equation  due  to  Kamerlingh 
Onnes  : 


where  A,  B,  C,  are  functions  of  temperature. 

-sm  .. 

Exercise  1.  —  If  I  -  —  \   =/is  the/rre  expansion  effect,  referring  to  adiabatic 
expansion  of  a  gas  into  a  vacuous  space,  show  that  : 


(A.  G.  Worthing,  1911.) 

83.     Liquefaction  of  Gases. 

The  cooling  on  free  expansion,  without  performance  of  external 
work,  is  made  use  of  on  a  large  scale  in  the  Linde  and  the  Hamp- 
son  apparatus  for  the  liquefaction  of  the  "  permanent  "  gases 
such  as  oxygen,  nitrogen,  and  air.  Strongly  compressed  gas  is 
allowed  to  escape  from  a  fine  nozzle,  and  the  cooled  gas  then 
sweeps  over  the  spiral  metal  tube  conveying  the  gas  to  the 
nozzle.  The  cooling  effect  is  thus  made  cumulative,  and  after  a 
sufficient  time  drops  of  liquid  are  ejected  from  the  nozzle. 

We  have  seen  that  hydrogen  becomes  slightly  warmed  in  this 
process,  so  that  its  liquefaction  by  free  expansion  would  be 
impossible  under  ordinary  conditions.  Dewar  in  1900  showed, 
however,  that  if  the  hydrogen  was  previously  cooled,  it  suffered  a 
further  cooling  on  free  expansion,  and  in  this  way  he  obtained 
liquid  hydrogen.  Olszewski  (1902)  found  that  the  inversion  point 
of  hydrogen  is  situated  at  —  80'5°  C.  This  effect  of  temperature  is 
general,  and  implies  that  the  ratio  of  the  potential  to  the  kinetic 


168  THERMODYNAMICS 

energy  of  gases  decreases  with  rise  of  temperature,  a  result 
which  is  quite  intelligible  if  we  suppose  the  molecular  forces  of 
attraction  or  repulsion  to  vary  inversely  as  some  power  of  the 
distance  between  the  molecules,  because  the  latter  must 
necessarily  separate  to  greater  distances  as  the  temperature 
increases  at  constant  pressure. 


CHAPTER  VII 

CHANGES    OF    PHYSICAL    STATE 

84.     Heterogeneous  Equilibria  ;  The  Phase  Rule. 

According  as  it  consists  of  one  or  of  more  phases,  a  system  in 
equilibrium  is  said  to  be  homogeneous  or  heterogeneous. 

If  a  heterogeneous  equilibrium  is  such  that  the  pressure  of 
the  system  depends  on  the  temperature  alone,  and  is  unchanged 
when  the  phases  alter  in  relative  amount,  it  is  called  a  completely 
heterogeneous  equilibrium  (Roozeboom),  or  an  indifferent  equili- 
brium (Duhem). 

We  shall  show  later  that  this  can  only  occur  when  the  phases 
are  of  unvarying  composition.  A  particular  case  is  a  pure  sub- 
stance in  different  states  of  aggregation. 

For  the  purpose  of  classifying  heterogeneous  equilibria  we 
shall  make  use  of  a  very  general  law,  called  the  Phase  Rule  of 
Willard  Gibbs  (1876),  the  proof  of  which  is  deferred  to  a  later 
chapter. 

This  is  an  equation  which  fixes  the  relation  existing  between 
the  number  of  phases  (?•),  the  number  of  components  (n),  and  the 
variance,  or  number  of  degrees  of  freedom  (F),  of  a  heterogeneous 
system  in  equilibrium,  subject  to  certain  conditions  which  are 
usually  satisfied  in  practice.  The  rule  states  that 
F  =  n  +  2  —  r. 

By  the  components  of  the  system  we  are  to  understand  the 
least  number  of  independently  variable  constituents,  in  terms  of 
which  the  composition  of  every  phase  in  the  system  can  be 
completely  specified.  The  number  of  components  will  therefore 
contribute  to  the  total  number  of  independent  variables  defining 
the  state  of  chemical  and  physical  equilibrium  of  the  system.  It 
is  not  necessary  that  the  components  shall  be  actual  constituents 
of  the  system ;  all  that  is  required  is  that  they  shall  be  inde- 
pendently variable,  i.e.,  the  least  number  has  been  chosen.  Thus, 
in  systems  composed  of  solid  fuming  sulphuric  acid  in  presence 


170  THERMODYNAMICS 

of  liquid  and  vapour,  we  might  conceivably  have,  under  various 
conditions,  the  following  phases  : 

solid :  S03,  H2S207,  H2S04,  H2S04.H20,  etc. ; 
liquid  :  H2S207,  H2S04,  or  mixtures  of  these  ; 
gaseous  :  H2S207,  H2S04,  S03,  or  mixtures  of  these. 

The  composition  of  every  possible  phase  could,  however,  be 
completely  specified  in  terms  of  its  content  of  S03  and  H20,  and 
these  two  may  be  taken  as  components,  although  neither  may 
really  exist,  as  such,  in  the  system. 

By  the  variance,  or  number  of  degrees  of  freedom  of  the 
system,  we  mean  the  number  of  independent  variables  which 
must  be  arbitrarily  fixed  before  the  state  of  equilibrium  is  com- 
pletely determined.  According  to  the  number  of  these,  we  have 
arariant,  univariant,  bivariant,  invariant,  .  .  .  systems.  Thus, 
a  completely  heterogeneous  system  is  univariant,  because  its 
equilibrium  is  completely  specified  by  fixing  a  single  variable — 
the  temperature.  But  a  salt  solution  requires  two  variables — 
temperature  and  composition — to  be  fixed  before  the  equilibrium 
is  determined,  since  the  vapour-pressure  depends  on  both. 

Heterogeneous  systems  may  differ  in  respect  of  the  number  of 
phases  and  their  state  of  aggregation  and  composition. 

I.  One  component : 

(i.)  One  phase  :  F  =  2,  i.e.,  a  fluid  (Chap.  V.)  ; 
(ii.)  Two  phases  :  F  =  1,  i.e.,  changes  of  physical  state  ; 
(iii.)  Three  phases  :    F  =  0,  i.e.,  the   phases   coexist  at   one 
definite  temperature  and  pressure  only  (triple point) . 

II.  Two  components : 

(i.)  One  phase  :  F  =  3,  i.e.,  a  fluid  of  varying  composition  ; 

(ii.)  Two  phases  :  F  =  2,  e.g.,  a  salt  solution  in  contact  with 
vapour  of  the  solvent,  or  frozen  solvent ; 

(iii.)  Three  phases  :  F  =  1,  i.e.,  the  equilibrium  is  completely 
heterogeneous,  e.g.,  CaC03  in  presence  of  CaO  and 
C02; 

(iv.)  Four  phases  :  F  =  0,  the  quadruple  point,  as  for  example 
chlorine  hydrate,  ice,  solution,  and  vapour,  or  the  eutectic 
point  of  a  salt  solution,  where  solid  salt,  solid  solvent, 
saturated  solution,  and  vapour  coexist  at  a  definite  tem- 
perature and  pressure.  A  peculiar  case  of  II.  (ii.)  is  that 
in  which  both  phases  have  always  the  same  composition 
(e.g.,  solid  NH4C1  and  the  vapour  composed  of  NH3  -f-  HCl ; 


CHANGES   OF   PHYSICAL    STATE 


171 


mixtures  of  maximum  and  minimum  boiling-point).  This 
may  be  treated  as  case  I.  (ii.) . 
III.  Three  components. 

The  interesting  case  is  F  =  0,  i.e.,  r  =  5,  a  quintuple  point  in 
which  five  phases  coexist.  An  example  is  the  system  formed  on 
heating  a  mixture  of  Glauber's  salt  (Na2S04 .  10H20)  and  Epsom 
salt  (MgS04 .  7H20)  to  22°,  when  partial  liquefaction  occurs  with 
formation  of  astracanite  Na2Mg(S04)2 .  4H20,  and  five  phases  are 
produced  : 

(Na2S04 . 10H20 

3  Solids  -:MgS04.7H20  saturated  solution,  vapour. 

(Na2Mg(S04)2.4H20 

85.     Evaporation. 

If  heat  is  supplied  to  a  liquid,  a  portion  of  the  liquid  on  the 
surface  passes  into  the  state  of  vapour.  If  the  liquid  is  freely 
exposed  to  air,  the  whole  gradually 
disappears  by  this  process  of  evaporation, 
but  if  it  is  contained  in  a  closed  vessel 
the  transition  into  vapour  is  limited,  the 
formation  of  the  latter  ceasing  when  it 
has  attained  a  certain  pressure,  called 
the  vapour-pressure  of  the  liquid.  Dalton 
(Mem.  Manchester  Phil.  Soc.  15, 409, 1801) 
established  the  following  laws,  which 
have  been  verified  by  later  observers : 

(1)  The    vapour-pressure    of   a    pure 
liquid  depends    on,  and  increases  with, 
the  temperature. 

(2)  It  is  independent  of  the  volume  of  the  vapour-space,  provided 
that  liquid  is  always  present. 

(3)  It    is   almost  independent  of   the  presence  of  indifferent 
gases  in  the  vapour-space  (Law  of  Partial  Pressures). 

These  statements  are  only  true  when  the  liquid  is  a  pure  sub- 
stance, i.e.,  does  not  change  in  composition  during  evaporation. 
This  constancy  of  vapour-pressure  serves  to  distinguish  pure  sub- 
stances from  solutions.  The  effects  of  surface  tension,  appearing 
when  small  droplets  are  used,  and  of  electrification,  must  also 
be  absent  (cf.  §§  100—102). 

Let  us  suppose  we  have  a  mass  of  pure  liquid  confined  over 


172 


THERMODYNAMICS 


mercury  in  a  tube,  and  that  we  measure  corresponding  pressures 
and  volumes  for  a  series  of  different  temperatures.  If  the 
isotherms  are  drawn  on  a  p,v  diagram  (Fig.  29),  each  is  seen  to 
consist,  in  general,  of  three  parts : 

(i.)  A  nearly  vertical  part  AB,  along  which  the  pressure  of  the 
homogeneous  liquid  is  diminishing  very  rapidly  with 
increase  of  volume.  At  B  the  homogeneous  liquid  sepa- 
rates into  a  heterogeneous  complex  of  liquid  and  vapour, 
the  curve  suddenly  changes  in  direction,  and  gives  : 
(ii.)  a  horizontal  line  BC,  called  the  line  of  heterogeneous  states, 
along  which  the  pressure  is  constant,  and  equal  to  the 
vapour-pressure  at  the  given  temperature.  When  the 
last  drop  of  liquid  has  evaporated,  the  curve  turns  sharply 
downwards  at  C,  giving  rise  to  : 

(iii.)  a  curve  CD,  indicating  the  compressibility  of  the  vapour. 
This  approaches  more  and  more  closely  to  a  rectangular 
hyperbola  with  falling  pressure. 

The  portion  BC  is  so  characteristic  that  this  procedure  gives  a 
very  accurate  method  of  measuring  vapour-pressures. 

86.     Critical  Phenomena. 

If  the  temperature  is  increased  still  further,  it  would  appear 


48-1 


probable  that  the  horizontal  line  of  heterogeneous  states,  which 
shrinks   rapidly   with   rise   of    temperature,    would    ultimately 


CHANGES  OF  PHYSICAL   STATE  173 

vanish  at  a  definite  point  P.  The  curve  would  now  slope  down- 
wards from  the  p  axis  at  every  part,  without  any  sharply 
defined  horizontal  portion,  i.e.,  there  should  be  no  liquefaction 
at  all.  This  remarkable  result  was,  in  fact,  obtained  by  Andrews 
in  1869  with  carbon  dioxide.  Being  impressed  with  a 
curious  observation  of  Thilorier's  (1835),  that  liquid  carbon 
dioxide  expands  on  warming  four  tunes  as  rapidly  as  the  gas 
between  0°  and  20°,  he  submitted  the  relations  between  the 
pressures  and  volumes  of  this  substance  at, different  tempera- 
tures to  a  very  careful  investigation.  The  gas  was  compressed  over 
mercury  in  a  strong  glass  tube,  and  the  (p,v)  isotherms  plotted 
for  13°'l,  21°-5,  31°'l,  31°'o,  33°'5,  48°'l  (Fig.  30).  Below  31°'3 
separation  into  a  heterogeneous  system  ("liquefaction")  began 
at  a  definite  pressure  for  each  temperature,  and  the  isotherms 
exhibited  flat  portions.  These  diminished  in  length  with  rise  of 
temperature,  and  at  31°*3  the  curve  simply  changed  its  direction 
at  a  pressure  of  75  atm.  and  showed  no  flat  part  at  all.  At  the 
same  time,  no  liquefaction  could  be  observed  in  the  tube,  no  matter 
how  high  the  pressure  was  taken.  Above  31C>3  the  isotherms 
gradually  lost  the  definite  change  of  direction,  which  may  be 
regarded  as  a  lingering  suggestion  of  the  flat  portion,  and 
approached  the  hyperbolic  form  at  48°'l. 

The  results  of  Andrews'  experiments  may  be  summarised  and 
generalised  in  the  statement  that : 

There  is,  for  every  gas  or  vapour,  a  definite  temperature,  above 
which  it  is  impossible,  by  increase  of  pressure  alone,  to  effect  a 
liquefaction  of  that  gas  or  vapour.  This  is  called  the  critical 
temperature  (0  ). 

The  critical  temperature  is  the  highest  temperature  at  which  a 
gas  may  be  liquefied  by  pressure,  and,  since  the  pressure  increases 
with  the  temperature,  there  will  correspond  to  the  critical  tem- 
perature a  critical pressure(pK),  which  is  the  greatest  pressure  which 
will  produce  liquefaction.  This  pressure  is  given  by  the  ordinate 
of  the  critical  point  K,  or  point  of  inflexion,  on  the  critical 
isotherm. 

The  volume  of  unit  mass  of  the  substance,  under  the  critical 
pressure  and  at  the  critical  temperature,  is  called  the  critical 
volume,  rK.  The  reciprocal  of  this  is  the  critical  density, 

Ps.  =  !/«*• 

A  substance  existing  at  its  critical  temperature  and  pressure 


174  THERMODYNAMICS 

must  of  course  have  the  critical  density ;  it  is  then  said  to  be  in 
the  critical  state,  (PK.,VK,  0K). 

The  ratio  #K/^K  =  k  is  called  the  critical  coefficient  of  a  substance 
(Guye). 

87.     Continuous   Transition   of  States. 

It  is  possible,  by  suitable  changes  of  pressure  and  temperature, 
to  pass  directly  from  a  point  a  in  the  vapour  region  to  a  point 

/3  in  the  liquid  region,  without 
any  discontinuity  in  the  way  by 
separation  into  two  phases.  The 
vapour  is  first  heated  above  the 
critical  temperature,  and  so 
brought  on  an  isotherm  AB 
(Fig.  31)  above  the  critical 
isotherm.  By  raising  the  pres- 
sure, the  state  passes  along  this 
isotherm  until  the  volume  is 
reduced  to  the  value  correspond- 

—  ing  to  a  point  ft  lying  on  a  liquid 

isotherm.     The    temperature    is 

now  lowered,  at  constant  volume,  till  it  reaches  the  value 
corresponding  to  the  isotherm  on  which  ft  lies.  If  the  vessel 
is  now  opened  it  is  found  to  be  completely  filled  with  homogeneous 
liquid.  There  are,  therefore,  two  ways  of  passing  from  vapour 
(a)  to  liquid  (/3),  or  rice  versa : 

(i.)  Along  aCD/3,  through  the  region  of  heterogeneous  states — a 

Heterogeneous,  or  Discontinuous,  Path. 
(ii.)  Along  aAB/3 — a  Homogeneous,  or  Continuous,  Path. 
The  dotted  curve  PQR,  separating  the  region  of  heterogeneous 
states   from   the  regions  of  homogeneous  states,  is  called  the 
limiting  carve  of  heterogeneous  states,  or  the  border  curve. 

This  conception  of  the  continuity  of  the  liquid  and  gaseous 
states,  expressed  by  Andrews  in  the  form  that  liquid  and  vapour 
are  "  only  distinct  stages  of  a  long  series  of  continuous  physical 
changes,"  is  the  basis  of  a  remarkable  theory  of  J.  D.  van  der 
Waals,  which  will  be  considered  later. 

Andrews  proposed  to  call  a  substance  existing  above  its  critical 
temperature,  a  gas,  as  distinguished  from  a  vapour,  existing  below 
that  temperature.  Whereas  vapours  may  be  liquefied  by  pressure 


CHANGES   OF   PHYSICAL   STATE 


175 


alone,  gases  require  cooling  below  the  critical  temperature  before 
they  can  be  compressed  to  liquids.  This  explains  the  failure 
of  Natterer's  attempts  to  liquefy  the  "permanent  gases": 
as  a  matter  of  fact  their  critical  temperatures  lie  below  the 
atmospheric  temperature. 

Above  the  critical  temperature,  the  isotherms  show  two  curva- 
tures ;  on  the  right,  pr  increases  as  p  decreases,  on  the  left,  pi- 
and  p  increase  together.  This  change  of  curvature,  suggesting 
vapour  and  liquid  states  respectively,  is  clearly  seen  in  Andrews' 
curves  for  32°'5  and  35°*5.  It  is  also  observed  in  the  isotherms 
of  permanent  gases,  and  will  be  discussed  later. 


88.     Thermodynamics  of  Evaporation. 

Let  a  quantity  of  liquid  and  its  vapour  be  in  equilibrium,  in  a 
cylinder  with  a  piston  under  a 
pressure  (p  —  fy>),  at  a  tempera- 
ture (T  —  ST). 

( P  —  &P)  *s  the  vapour-pressure 
at  (T  —  8T)°. 

Let  PR,  QS  (Fig.  32)  be  the 
isotherms  (T  —  8T)  and  T,  where 
P,Q  correspond  to  pressures 
(p  —  fy>)  and  p.  PR,  QS  are 
horizontal,  since  p  is  a  function 
of  T  only,  and  does  not  depend 
on  the  volume. 

We  now  take  the  system  round 
a  reversible  Carnot  cycle,   the 
reversibility  for  such  a  system,  in  the  limit,  having  been  pre- 
viously demonstrated. 

(1)  Compress   adiabatically  along    AB   till    the    temperature 
reaches  T. 

(2)  Transfer  to  the  hot  reservoir  and  expand  isothermally  till 
a  mass  in  of  liquid  has  been  evaporated.     The  heat  absorbed  along 
BC  is  wLe  where  Le  =  latent  heat  of  evaporation  at  T°. 

(3)  Transfer  to  the  non-conducting  stand,  and  expand  adia- 
batically along  CD  till  the  temperature  falls  to  (T  —  ST). 

(4)  Transfer  to  the  cold  reservoir,  and  compress  isothermally 
along  DA  till  the  initial  state  is  reached. 


FIG.  32. 


176  THERMODYNAMICS 

The  adiabatics  are  small  curve-elements,  which,  for  an  infinite- 
simal cycle,  may  be  regarded  as  parallel  straight  lines.  Draw 
BF,  CE  perpendicular  to  the  volume  axis  Or. 

Work  done  in  cycle  =  area  of  parallelogram  ABCD 

=  area  of  rectangle  FBCE  =  FB  .  BC. 

But  FB  =  increase  of  pressure  at  constant  volume  due  to  rise 

of  temperature  8T  =  -~r  ST,  and 

BC  =  change  of  volume  due  to  evaporation  of  in   gr.   of 

liquid 
=  (volume  of  m  gr.  of  vapour)  —  (volume  of  m  gr. 

liquid) 

=  mi's  —  mi'i  =  in(i'z  —  i'i), 

where  i\,  r2  are  the  specific  volumes  of  liquid  and  vapour  at  T°, 
respectively. 

Hence,  work  done  per  cycle  =  -^  8T(r2  —  i'i)nt. 

By  Carnot's  theorem,  and  the  definition  of  absolute  tempera- 
ture, this  magnitude  =  Q  X  -m-  ,  where  Q  =  heat  absorbed  from 
the  hot  reservoir, 

5T 
=  mLe  X  j 


This  equation  is  known  as  the  Clapeyron-Clausius  equation. 
(Clapeyron  1834  ;  Clausius  1851.) 

We  need  not  write  this  (  ~ I ,  as  would  appear  at  first  sight,  since  p 

is  a  function  of  T  alone,  and  is  independent  of  the  volume  so  long  as  both 
phases  are  present.  A  given  rise  of  temperature  therefore  produces  the 
same  change  of  vapour-pressure  whether  the  volume  is  kept  constant  or  not. 

Le  =:  latent  heat  of  evaporation. 
t'2  —  vi  EE  Ar  =  volume  change  accompanying  unit  mass  of 

phase  transition  at  the  pressure  p. 
T  =  temperature  at  which  the  phases  are  in  equilibrium 

under  the  vapour-pressure  p. 
Corollary  1. — Since  Ar,  Le  are  known  by  experiment   to  be 


CHANGES  OF  PHYSICAL   STATE  177 

positive,  the  vapour-pressure  of  a  liquid  always  increases  with 
rise  of  temperature. 

Corollary  2. — The  increase  of  vapour-pressure  for  a  very  small 
change  of  temperature  is  proportional  to  the  change  of  tempera- 
ture ;  also  the  rise  of  boiling-point  of  a  liquid  for  a  very  small 
increase  of  external  pressure  is  proportional  to  the  increase  of 
pressure. 

The  boiling-point  of  a  liquid  is  the  temperature  at  which  its  vapour-pres- 
sure is  equal  to  the  total  pressure  to  which  the  liquid  is  subjected,  usually 
the  atmospheric  pressure,  and  a  liquid  begins  to  boil,  i.e.,  emits  bubbles  of 
vapour,  when  its  temperature  is  raised  to  the  point  at  which  its  vapour- 
pressure  is  equal  to  the  total  pressure  on  the  surface.  The  bubbles  of  vapour 
are  always  evolved  at  definite  points,  where  small  gas-bubbles  (either 
evolved  from  the  liquid,  or  due  to  air  adhering  to  the  vessel)  are  present. 
As  boiling  proceeds,  these  points  diminish  in  number,  owing  to  expulsion 
of  gas,  and  after  a  sufficient  time  disappear  altogether.  The  temperature 
then  rises  several  degrees  above  the  boiling-point  without  formation  of 
bubbles ;  at  last,  however,  an  explosive  rush  of  vapour  is  evolved,  usually 
in  one  large  bubble  from  the  bottom  of  the  vessel,  the  temperature  sinking 
again  to  the  boiling-point,  and  the  process  is  repeated.  According  to 
Aitken  (1874),  normal  boiling  occurs  only  if  bubbles  or  cavities,  and  hence 
the  vapour  of  the  liquid,  are  present.  This  result  is  quite  general ;  the 
phase-transition  proceeds  normally  only  if  both  phases  are  present ;  if  only 
one  phase  is  present,  the  transformation  into  the  other  lags  behind  the 
change  of  temperature  or  pressure,  and  then  occurs  violently  :  the 
phenomenon  is  called  Suspended  Transformation  of  States. 

If  the  vapour  of  the  liquid  is  allowed  to  form  inside  a  closed 
vessel,  its  pressure  continually  increases  with  rise  of  temperature, 
the  maximum  vapour-pressure  being  the  critical  pressure.  Con- 
versely the  boiling-point  is  raised  if  the  pressure  is  increased  by 
enclosing  the  liquid  and  vapour  in  a  vessel  fitted  with  a  safety-valve, 
or  by  increasing  the  external  (e.g.,  atmospheric)  pressure.  The  rise 
of  boiling-point  was  first  described  by  the  French  inventor  Denis 
Papin  (1679)  and  applied  to  the  construction  of  a  "  new  digester, 
or  engine  for  softening  bones  " — now  called  an  autoclave.  The  rise 

of  boiling-point  is  calculated  from  the  formula  oT  =  -==—•  ty. 

Example. — Evaporation  of  water  at  100°  under  1  atm.  pressure  : 

T  =  boiling-point  =  273  +  100  --=373, 

vi  =  sp.  vol.  of  saturated  vapour  =  1,674  c.c. 

v%  =  sp.  vol.  of  liquid  =  1  c.c.  (nearly) 

dP  _o-i.)  mm.Hg    ,  n   _  27'12  X  1,013,250  dyne  per  om.g 

dT~          '     1°  0.     *  760  1° 

T.  N 


178  THERMODYNAMICS 

_  ;n;j  x  1,073  x  27-12  x  1,013,250       _  538.8  cal  /obs  -38.7^ 

760 

If  we  refer  the  latent  heat  to  a  mol  of  liquid,  instead  of  unit 
mass,  we  have,  if  MI,  M2  are  the  molecular  weights  of  liquid  and 
vapour : 

Vi  =  MI?'I  ;  V2  =  M2r2,  the  molecular  volumes, 
\e  =  MiL,,,  the  molecular  heat  of  evaporation, 

dp  _  ^e  .         .         .     (2) 


Tf  M  _  M  .     P  —  __A__  .  (2o) 

Ma  '  dT       T(V2  -  Vi) 

Corollary  1.  —  Since  Vi  is  very  small  compared  with  Y2,  we  may 
neglect  it  as  a  first  approximation,  and  write  : 


where  V  =  molecular  volume  of  the  vapour  at  (p,  T). 

Corollary  2.  —  If  the  saturated  vapour  obe}7s  the  gas  laws  : 
V  =  BT/p, 


.    d^_  Xep 
'  dT  ~  RT2 


1  dp        dln  . 

or       =      =         •     •     • 


The  condition  assumed  is  not  very  approximately  true  unless 
we  are  dealing  with  a  substance  of  very  small  vapour-pressure 
(e.g.,  mercury  at  15°  has  a  vapour-pressure  of  O'OOOSl  mm. 
according  to  Pfaundler  (1897) ).  It  will  therefore  apply  more 
particularly  to  low  temperatures. 
Example. — In  the  case  of  water  : 

R  =  1-985  cal. 

p  =  760  mm. 

T  =  373 

dp       nf_       mm. 

3f  =  27'1!  -i5" 

T         Ac        1-985  X3732  X  27-12 

•'•  Le  =  M  =         ~Wx^8-        =  547'°  Cal" 

which  is  about  2  per  cent,  too  large.     The  molecular  volume  of 

saturated  water  vapour  at  100°  is  less  than  that  calculated  from 

the  gas  laws  (Fairbairn  and  Tate,  1860  ;  Clausius,  1861,  showed 


CHANGES  OF   PHYSICAL   STATE  179 

that  the  values  for  r2  obtained  by  these  observers  agreed  very 
well  with  those  calculated  from  the  latent-heat  equation).  l/ra  is 
called  the  density  of  saturated  vapour. 

The  methods  used  for  the  determination  of  the  density  of  a  saturated  vapour 
will  be  found  in  the  treatises  on  Physics  (cf.  Young,  Zeitschr.  Physical.  Chem., 
70,  620,  1910,  who  also  finds  the  very  simple  relation  : 


where  p  =  vapour-pressure;  A,  B  =  constants). 
Corollary  3.  —  By  transposing  the  terms  of 
dln  Ae 


we  obtain 


According  to  Clausius  the  latent-heat  of  evaporation  of  a  liquid 
is   approximately   a   linear  function   of    the   temperature,   and 
diminishes  with  rise  of  temperature  (cf.  §  94): 
Xe  =  A0  —  aT, 


which  gives  on  integration 

~R 


hip  =  —  ~  —  :p  InT  +  constant, 


where  A,  B,  C  are  constants  for  a  particular  substance.     This 
equation  is  due  to  Kirchhoff  (Pogg.  Ann.  104,  612,  1858.) 

The  equation  is  usually  called  Dupre's,  or  Rankine's,  formula. 
The  latter  name  might  well  be  adopted  for  the  equation  : 

£          7          § 
-- 


(Rankine,  Edin.  Phil.  Journ.  1849),  recently  put  forward  again 
by  E.  Bose  (Physik.  Zeitschr.  8,  944,  1907). 

At  least  thirty  formulae  representing  the  vapour-pressure  of  a 
liquid  (usually  water)  as  a  function  of  temperature  have  appeared, 
and  new  ones  are  always  being  published.  Some  of  the  best 
known  are  due  to  : 

(1)  Biot  (1844)  :  log  p  =  a  +  bar  +  c,8T 
a,  b,  a,  /3  are  constants, 
I*  =  6°  C.  —  const. 

N  2 


180  THERMODYNAMICS 

(2)  Magnus  (1844)  :  p  =  ab  t+e 

rnn 

(3)  Bertrand  (1887)  :   p  =      "\_    ,m 

(4)  Van  der  Waals  (1899)  :   log  p  =  —  a  ^K  +  a  +  log  j>K 

where  TK,  pK  are  the  critical  constants. 

(Cf.  Winkelmann  :   Plnjsik.,  III.,  3,  903-961.) 

The  nature  of  the  assumptions  entertained  in  the  deduction  of 
KirchhofFs  equation  ensure  the  theoretical  validity  of  the  latter 
only  at  low  temperatures.  Notwithstanding  this,  Juliusburger 
(1900),  from  a  review  of  existing  data,  found  the  equation  to  give 
results  deviating  by  3  per  cent,  at  most  from  the  observed  values 
for  74  substances  out  of  80  examined.  He  considers  that  it  may 
be  regarded  as  an  empirical  formula  with  three  constants,  up  to 
the  critical  point,  and  gives  the  values  of  A,  B,  C  for  a  number 
of  substances. 

(P.  Juliusburger,  Drude's  Ann.  3,  618,  1900.) 

89.     Empirical  Relations. 

Interpolation  and  extrapolation  of  the  vapour-pressure  curve 


could  be  carried  out  by  means  of  successive  applications  of  the 
Clausius   equation  were   it  not  that  data   are  usually  lacking. 
Eamsay  and  Young  (1886)  discovered  an  empirical  rule  by  which 
the  interpolation  can  be  effected.     Let 
TA,  TA'  be  the  absolute  boiling-points  of  a  substance  A  under 

pressures  p  and  p', 
TB,  TB'  the  boiling-points  of  another  substance  B  under  the  same 

pressures  p  and  p', 

then  |4  =  ^  +  c  (TB'  -  TB) 

IB         IB 
where  c  is  a  constant,  usually  very  small. 

In  the  case  of  substances  which  are  chemicall    similar 


rr    i  rp  m 

m^ FTT  =  rT^  =  constant, 


T  '       T 

and  CftHgBr,  esters,  aromatic  substances)  c  =  0  and  -f~~  =  7=^. 

IB         IB 

From  this  we  easily  find 
T/  -  TA 
TB'  —  TB 
a  rule  which  was  put  forward  by  Diihring  in  1878. 


CHANGES   OF  PHYSICAL   STATE  181 

If  in  the  equation  ^  =  ^ — - — r  we  put  (j-2  —  i'i)  =  0  and 
al       I(r2 — 1\) 

L€  =  0,  simultaneously,  the  gradient  of  the  vapour-pressure 
curve  becomes  indeterminate,  and  another  condition  must  be 
given  to  fix  the  latter,  viz.,  the  mass  of  substance.  This  was 
verified  by  Cailletet  and  Colardeau  (1891),  who  found  that  the 
vapour-pressure  curves  for  various  quantities  of  liquid  fell 
together  up  to  the  critical  point,  but  then  diverged  into  a  bundle 
of  curves. 

Of  the  two  conditions : 


the  first  is  seen  to  be  satisfied,  at  the  critical  point,  from  an 
inspection  of  Andrews'  diagram,  and  the  second  is  also  verified 
by  experiments  of  Mathias  (1890). 

Both  conditions  must  be  satisfied  simultaneously,  for  the  con- 
dition that  both  phases  become  identical  at  the  critical  point 
certainly  requires  that  their  intrinsic  energies  and  entropies  per 
unit  mass  are  equal : 

w2  —  Ml  =  0,  s2  —  si  =  0     .         .         .     (i.) 
But  ?/2  —  MI  =  Le  —  Xl'a  —  i'i)  •        •        •    (ii.) 

,                                            Le  ,...  ^ 

and  s2  —  si  =  -^ (in.) 

Since  T  is  finite,  «2  —  «i  can  vanish  only  when  the  two  con- 
ditions (a)  are  simultaneously  satisfied. 

If  the  phase  transition  is  of  such  a  kind  that  either  condition 
can  be  satisfied  separately,  but  not  both  together,  it  cannot 
exhibit  a  critical  point  (Tammann). 

90.     Metastable  States. 

States  such  as  superheated  liquid  and  supercooled  vapour  are 
known  as  metastable,  they  are  not  of  themselves  unstable,  but 
become  so  on  introduction  of  a  small  amount  of  the  stable  phase. 

Metastable  states  are  represented  on  the  indicator  diagram  by 
the  prolongations  of  the  isotherms  of  homogeneous  states  beyond 
the  intersections  with  the  line  of  heterogeneous  states.  Thus,  the 
productions  of  the  liquid  and  vapour  portions  of  the  0:  isotherm, 
Fig.  33,  meet  the  heterogeneous  parts  of  the  isotherms  02,  #3  at 
Q,  P,  respectively.  At  each  of  these  points  there  are  tico  con- 
ditions of  existence  possible  for  the  system,  thus : 


182 


THERMODYNAMICS 


P  corresponds  (a)  to  a  stable  heterogeneous  system  on  the  #3 
isotherm  at  P ;  (b)  to  a  metastable  homogeneous  state  on  the  QI 
isotherm  produced  to  P  (#3  >  #1). 

Q  corresponds  to  (a)  a  stable  heterogeneous  system  on  the  02 
isotherm,  (b)  a  metastable  homogeneous  state  on  the  production 
of  the  61  isotherm  (0*  <  #1). 

James  Thomson  (1871)  suggested  that  these  prolongations  of 
the  di  isotherm  of  a  liquid  and  vapour  might  ultimately  bend 
round  and  join,  so  that  the  discontinuous  flat  part  of  the  isotherm 
is  replaced  by  a  continuous  curve  (Fig.  34).  a/3  corresponds  to 


FIG.  33. 


FIG.  34. 


superheated  liquid,  78  to  supercooled  vapour ;  p<y  seems  to  be 
physically  unreal,  for  on  it  we  should  have  _p  and  v  increasing 
together,  a/3  may  pass  below  the  v  axis,  since  liquids  may  be 
exposed  to  a  tension  ,•  mercury  in  a  clean  barometer  tube  does 
not  always  fall  to  the  barometric  height,  but  remains  filling  the 
tube  until  the  latter  has  been  tapped  or  shaken.  0.  E.  Meyer 
has  in  this  way  exposed  ether  to  a  tension  of  —  70  atm.,  and 
Worthington  (1892)  has  shown  directly  that  the  compressibility 
of  alcohol  is  the  same  under  tension  as  under  pressure. 
The  curve  aft  is  therefore  continuous  where  it  crosses  the 
v  axis. 

The  line  a5  is  that  of  the  heterogeneous  states  ;  its  position 
is  fixed  by  the  following  rule,  due  to  Maxwell  (1875). 

If  we  suppose  the  substance  carried  round  a  complete  iso- 
thermal cycle  aftjbCa,  the  work  done  is  zero.  But  this  work  is 


CHANGES   OF   PHYSICAL   STATE  183 

represented  by  the  area  of  the  loop  =  —  (area  a/3Ca)  -f  (area 


.*.  area  affCa  =  area  CySc, 

so  that  the  line  ab  must  cut  the  isotherm  so  that  the  areas  inter- 
cepted above  and  Mow  it  are  equal  (cf.  Duhem  :  Mecanique 
chimiqtie,  ti.,  184). 

91.     Energy  and  Entropy  of  Saturated  Vapour. 

Let  iii,  u2  be  the  intrinsic  energies, 
Si,  s2  the  entropies, 
Vi,  r2  the  volumes, 

of  unit  mass  of  liquid  and  vapour,  respectively,  at  a  given  tem- 
perature and  pressure. 

If  Le  is  the  latent  heat  of  evaporation 

«a  —  "i  =  L,  —  X<2  —  ^i)      •         •         •     (1) 

«2  —  «i  =  -rjf  •         •         •         •        ••         •     (2) 

(u$  —  «i)  is  usually  called  the  internal  heat  of  evaporation,  and 
denoted  by  p.  All  these  magnitudes  are,  for  a  saturated  complex, 
functions  of  temperature  alone,  since  the  vapour-pressure  is 
definite  at  a  given  temperature. 

From  the  equation  (1)  of  §  88  we  find 


For  water,  adopting  the  values  of  §  88,  we  find  for  the  ratio 


so  that  the  external  work  is  very  small  in  comparison  with  the 
heat  absorbed. 

The  entropy  of  water  vapour  is  usually  referred  to  that  of 
liquid  at  0°  C.  as  zero,  and  the  entropy  of  unit  mass  of  steam  at 
6°  C.  is  therefore  : 


, + r  aw 

Jo 


11 


(9  +  273      ~  B  +  273 

where  L,,  is  the  latent  heat  of  evaporation  at  0°,  I   cd9  is  the 
heat  absorbed  in  raising  the  temperature  of  the  liquid  from  0°  to 


184 


THERMODYNAMICS 


the  temperature  of  evaporation,  and  H  is  the  total  amount  of 

heat  absorbed  in  the  change.     H  is  usually  called  the  total  heat 

of  the  vapour.  In  the  case  of 
steam,  Watt  (1781)  surmised  that 
H  is  independent  of  temperature, 
whilst  Southern  and  Creighton 
(1803)  thought  this  was  probably 
the  case  with  Lf.  Eegnault 
(1845)  by  experiment,  and  Clausius 
(1855)  by  calculation,  proved  that 
neither  quantity  is  constant,  but 
both  are  functions  of  temperature. 
The  entropy  and  intrinsic  energy 
of  steam  (or  other  vapours)  may 
be  determined  by  the  following 
method,  due  to  Clausius. 
Let  unit  mass  of  the  saturated  complex  be  taken,  consisting  of 

m  gr.  vapour  and  1  —  m  gr.  liquid. 

Let  the  initial  state  of  the  liquid  be  represented  by  A  on  the 

border  curve  (Fig.  35).      Let  the  temperature  at  A  be  T0. 
The  increase  of  entropy  on  passing  to  liquid  at  B(T)  is : 


FIG.  35. 


i 


T 


where  a'  is  the  specific  heat  of  the  liquid  in  contact  with  its 
saturated  vapour,  but  may  be  taken  as  the  ordinary  specific  heat 
at  constant  pressure. 

The   increase   of  entropy  on   passing  to  the  given  complex 
represented  by  the  point  M  on  the  line  of  heterogeneous  states  is  : 


/.  entropy  in  the  fctate  s  M  =  s  = 


ff 
"  °  J, 


(ra  — 


(1) 


(2) 


where  *0  is  the  entropy  in  the  standard  state  A. 

If  a  mixture  of  liquid  and  saturated  vapour  is  reversibly  com- 
pressed in  a  vessel  impervious  to  heat,  the  corresponding  values 


CHANGES   OF   PHYSICAL   STATE  185 

of  v  and  p  trace  out  an  adiabatic  line,  MM',  the  equation  of 
which  is  : 

s  =  const. 


If  M"  lies  on  another  adiabatic,  the  entropy  of  the  system  at 
M"  is 


The  heat  of  the  path  ABM  is  : 

Q  =  j      crV/T  +  IvH.         .         .         .     (5) 

J    T 

The  work  of  the  path  ABM  is : 

f'B  ; 
A  =       pdvi  -f-  p(r  —  i?j)       .         .         .     (6) 


where   ri  =   specific   volume   of   liquid,  r  =  total   volume   of 
complex,  at  the  temperature  T. 

The  increase  of  energy  in  passing  from  A  to  M 

=  AU  =  Q  —  A  =  T  aV/T  —  P  V/ri  +  Lfm  —  p(v  —  n)  .     (7) 
J  TO  J  rA 

But  TI  is  nearly  independent  of  p, 

.:  rfi-i  =  ^  rfT  =  aV/T 

Also  i?  r=  (1  —  ni)n  +  ?nr2  .  .  .  .  (8) 

where  r.2  =  specific  volume  of  the  vapour. 

Corollary.  —  The  point  M  is  the  centre  of  gravity  of  weights 
1  —  m  and  m  placed  at  the  liquid  and  vapour  ends  of  the  line  of 
heterogeneous  states. 

Thus  v  —  ri  =  w(r2  —  TI)  =  —  -~ 


m   "- 

rfT 


(9) 


f" 

=  pV  —  PQVQ  —  l'i 

J    pn 


186  THERMODYNAMICS 

We  integrate  (6)  by  parts  : 

A 
where  j>o,*'o  refer  to  A,  and  p,v  to  M.     Thus  : 

u  =  MO  +  m^e  —  (lyv  —  PQI'O)  + 
The  change  of  U  on  passing  to  any  other  state  M'  is  : 
u'  —  u  =  m'Le  —  mLe  —  (p'v'  —  pv)  + 


.     (6') 


^nHT   .    (9') 


il  1  \ 

v  +?\7rrT-  (9"} 

J    T 


Corollary. — If  M,  M'  are  on  the  same  adiabatic,  the  external 
work  done  is  given  by  (9"). 

92.     Specific  Heat  of  Saturated  Vapour. 

Clausius  (1850),  in  considering  Regnault's  data  for  the  latent 
heat  of  steam,  introduced  a  new  specific  heat,  applicable  to  either 
phase  of  a  saturated  complex  of  two  phases, 
viz.,  the  amount  of  heat  absorbed  in  raising 
the  temperature  of  unit  mass  of  a  saturated 
phase  by  1°,  the  pressure  being  at  the  same 
time  varied  so  as  to  preserve  the  substance 
in  a  saturated  state.  In  the  case  of  a 
vapour,  this  is  called  the  specific  heat  of 
saturated  vapour  (<r). 


FIG.  36. 


Thus,  if  we  consider  unit  mass  of  saturated 
steam,  in  equilibrium  with  liquid  (Fig.  36), 
to  be  isolated  without  change  of  temperature  and  pressure,  and  then 
to  be  heated,  this  vapour  would  become  unsaturated,  i.e.,  would  take  up 
more  liquid  at  its  own  temperature  if  this  were  offered  to  it.  To  prevent 
the  assumption  of  this  unsaturated  condition  we  must  compress  the  vapour 
during  the  addition  of  heat  so  that  the  pressure  is,  at  every  stage  of  the 
process,  equal  to  the  vapour-pressure  at  the  corresponding  temperature. 
The  heating  and  compression  of  the  vapour  may  then  be  performed  in 
contact  with  the  liquid,  so  that : 

(i.)  The  vapour  remains  saturated,  and  hence, 

(ii.)  The  masses  of  liquid  and  vapour  are  unchanged. 

Let  e,,"  =  specific  heat  of  vapour  at  constant  pressure, 
a"  =  specific  heat  of  saturated  vapour, 


CHANGES   OF  PHYSICAL   STATE  187 

then  by  definition,  and  equation  (2),  §  64: 

n  _  „  n    i    7    dP  _  „  „ 

-C»    +/^7T-^- 
Similarly,  for  the  liquid,  we  have  : 


These  equations  are  due  to  Clausius,  and  give  the  relation 
between  the  specific  heats  of  liquid  and  vapour  in  the  saturated 
complex  and  the  ordinary  specific  heats  at  constant  pressure. 


Since     ^m      is  very  small,  we  can  write,  approximately  : 

"•'  =  cp'. 

Let  unit  mass  of  saturated  complex,  consisting  of  m  parts  of 
vapour  and  (1  —  m)  parts  of  liquid,  be  contained  in  a  vessel,  and 
by  a  rise  of  temperature  let  a  further  small  quantity  dm  of  vapour 
be  produced.  The  pressure  at  the  same  time  rises  so  as  to 
preserve  the  state  of  saturation.  The  heat  absorbed  is  : 

bQ  =  Ledm-+{<r'(l-m)  +  o"m}dT!      .         .     (2) 
Hence  for  the  entropy  changes  we  have  : 


(-4) 


ds\ 
8T/«~~ 


T 
But  ds  is  a  perfect  differential,  hence 


8T8m 

Uite 


"  dm\  T  )"~f/TVT 

.  dLe  ,      L,, 

•  •    JTT  =   °^    —  °~    +   ST  •  •  •        (5) 


or  c/'-cr'^T  .         .  .  (6a) 

Equation  (5)  was  deduced  simultaneously  and  independently 
by  Clausius  and  Rankine  (1850). 
In  the  case  of  water  at  100°  C. 

<r'  =  1-01 ;  L,  =  538-7 ; 

^  =  -  0-61 ;  T  =  373  ; 
.-.  a"  =  1-01  —  0-61  —  1-44  =  —  1-04 


188  THERMODYNAMICS 

Hence  at  this  temperature,  we  arrive  at  the  somewhat  surprising 
result  that  the  specific  heat  of  saturated  steam  is  negative. 

The  specific  heats  of  the  saturated  vapours  of  €82,  CHC18,  CC14,  and  acetone 
were  also  found  to  be  negative  ;  that  of  ether,  however,  is  positive.  This 
remarkable  result  is  explained  by  the  fact  that  the  specific  heat  is  supposed 
to  be  measured  whilst  the  vapour  is  kept  saturated,  i.e.,  if  the  vapour  and 
liquid  are  heated  together,  no  liquid  must  evaporate.  To  prevent  liquid 
evaporating  as  the  temperature  is  raised  through  1°,  the  vapour  must  be 
compressed,  and  the  heat  evolved  from  the  work  of  compression  is  greater 
than  that  absorbed  in  raising  the  temperature,  .'.  a"  is  negative.  G.  A. 
Him  (1863)  verified  this  explanation  by  enclosing  saturated  water  vapour 
at  high  pressure  in  a  cylindrical  vessel  with  plate-glass  ends.  On  suddenly 
opening  a  tap,  and  so  expanding  the  vapour,  a  cloud  of  small  drops  formed 
in  the  cylinder.  With  ether,  however,  the  cloud  was  produced  when  the 
saturated  vapour  was  compressed  by  a  piston. 

Since  the  latent  heat  of  evaporation  always  decreases,  the  value  of  a"  for 
all  substances  increases  (algebraically),  with  rise  of  temperature  ;  if  negative, 
then  at  a  certain  temperature  it  becomes  zero,  and  then  positive.  Thus, 
above  127°  chloroform  shows  the  phenomenon  of  compressive  cloud-formation 
observed  with  ether. 

Mathias  (1896)  found,  in  his  experiments*  on  the  latent  heat  of  evaporation 
of  liquid  CO2,  S02,  and  N20  up  to  the  critical  points,  that  the  curve  Le  =/(T) 
cuts  the  T  axis  at  right  angles  at  the  point  T  =  TK.  Thence  at  the  critical 
point 


-'-'+ 

.•.  at  the  critical  temperature  a-"  =  —  <x>. 

Now  at  temperatures  considerably  less  than  TK,  <r"  increases  with  T, 
hence  at  some  intermediate  temperature  it  must  pass  through  a  maximum. 
If  there  is  also  a  point  of  inversion  from  —  to  -f-  at  lower  temperatures  (e.g., 
chloroform  at  127°)  there  are  therefore  two  points  of  inversion  for  «r"  ;  if  there 
is  no  such  point  there  is  probably  always  the  one  at  the  critical  temperature. 

Examples.  —  (1)  An  adiabatic  change  will  cause  an  increase,  or  decrease,  in 
the  quantity  of  vapour,  according  as  : 

<r'(l  —  m)  +  ff"ra^(). 

(2)  On  adiabatic  expansion  of  liquid  and  vapour,  show  that  (if  a'  is 
practically  constant)  : 

m'L/       mLe         I7    T' 
=—  ii  --  ff"n     - 


(3)  A  mixture  of  equal  weights  of  water  and  steam  at  100°  is  expanded 
adiabatically  in  a  cylinder.  Prove  that  some  water  will  evaporate. 

*  The  same  author  has  worked  out  aii  ingenious  method  for  the  deter- 
mination of  a',  tr",  L  as  far  as  the  critical  point  (Ctmttes  Rendus,  1894,  119 
404,  846). 


CHANGES   OF   PHYSICAL   STATE  189 

93.    Planck's  Equations. 

From  the  equations  of  the  previous  section  : 


-  -*--       •  <») 

together  with  the  latent-heat  equation  : 


we  readily  obtain  the  relations  : 


=v-v+?~*3|.*^ 

We  observe  that  the  differential  coefficient  -^  is  ?ota/,  i.e..  >> 

nl 

changes  along  with  T  so  as  to  maintain  saturation. 

To  find  the  value  of  (-=-e.  i.e..  the  rate  of  change  of  the  latent 
dp 

heat  with  the  saturation  pressure,  we  make  use  of  the  rule  for 
change  of  independent  variable. 


--.rfT=^         ....     (7) 

dL^_dLe     dp 
'  TlT      "dp   '  dT 

.    dL<     ^=a»  _a>  +  Lf  (8) 

'    dp  '  dT  T 

/.  from  (1)  and  (4)  : 

f-w-if+^-o  -•*£*>     .  w 

If  the  change  of  pressure  occurred  at  constant  temperature, 

dT 

=  0,  hence: 


Equations  (6)  and  (9)  are  due  to  Planck  (1897). 


190  THEEMODYNAMICS 

In  the  case  of  water  and  steam  at  100°  C. 
c  '  =  I'Ol ;  ~  =  -  0-610  ;  v"  =  1674  ;    ( ~  )    =  4'813 


t> 
~Le  =  538-7 ;    v'  —  I'O  ;  (^\    —  O'OOl ; 

thence  cp"  —  cp'  =  —  0'51 

.-.  Cp"  =  0-50. 

The  mean  specific  heat  of  saturated  steam  at  a  temperature 
slightly  higher  than  100°  was  found  by  Eegnault  to  be  0'48. 
If  the  saturated  vapour  is  assumed  to  obey  the  gas  laws 
*»  =  |     5,       (W 

rfLe 

Thus,  cj;"  =  I'Ol  —  0'61  =  0'40,  which  is  considerably  too  small.* 

94.     Kirchhoff's  Vapour-Pressure   Equation. 

In  the  case  of  the  evaporation  of  a  pure  liquid  : 

P=Le-p(v"-v')          .         .         .     (1) 

where  p  =  u"  —  u'  ....     (2) 

For  two  different  temperatures  TI,  T2, 

P%  —  Pi  =  W  —  Uz  —  (iii"  —  MI}- 
If  we  assume  the  specific  heat  of  the  vapour  to  be  constant : 

u"  =  wo"  +  c/T         .         .         .         .  (3) 

*       -?/  ""    /)/    ff   ^   ft  /T~1  rp   \  /O      \ 

Assume  for  the  liquid  an  expression  of  the  form  : 

f   f    f    /m  TT\    \  //i  \ 

where  c'  is  a  mean  specific  heat  between  T2  and  TI, 

.-.  P2  -  Pl  =  (Ta  -  TO  (cp"  -  c')   .         .         .  (5) 
Put  p!  —  T!  (cv"  —  c')  =  c 

and  Pa  =  p,  T2  =  T, 

.'.  p  =  Le  —p(v"  —  v'}  =  c  +  (c,"  —  c')T  .         .     (6) 

.'.  also  T(v"  —  v')  ^  —  p(v"  —  v')  =  c  +  (c,"  —  c')T. 

If  we  neglect «/'  in  comparison  with  v",  and  assume  the  vapour 
obeys  the  gas  laws,  we  have,  for  a  mol : 

ET2^     -r=ET  +  C  +  (C;'-C')T     .        .     (7) 

*  (Cf.  Callendar,  Proc.  Roy.  Soc.,  67,  266,  1901;  Dieteiici,^ln».PAy«.,  13, 
154,  1903.) 


CHANGES   OF   PHYSICAL    STATE  191 

Put  C/R  =  B',  jj((V'-C')  =  7 

.    l     ^_I   ,   B'    ,  T 

•  p  •  dT  ~  T  "T"  f*  "T  T 

.         .     (8) 


or  log  10  p  =  A'  -      +  C  log  MT  .         .         .  (8«) 

This  is  Kirchhoff's  vapour-pressure  equation  (§  88). 
It  may  be  written  in  the  form  (Hertz,  (1882)  )  : 

i^-rV  •  w 

where  o,  A  are  constants. 

It  was  stated  (I.e.}  that  (8)  gives  very  accurate  results  with  empirical  values 
of  7,  A,  and  B.  Bertrand  found,  however,  that  these  values  do  not  even 
approximately  agree  with  those  calculated  from  A  =  C/R,  7  =  (  €„"—  C')/E, 
except  in  the  case  of  steam. 

Graetz  (1903)  has  proposed  a  slight  modification  of  Kirchhoff's  equation  : 


.         .         .     (10 

-i.  JL 

where  k  is  another  constant. 

95.     Sublimation. 

Many  solid  substances  (camphor,  iodine,  naphthalene,  etc.),  are 
known  which  are  appreciably  volatile  at  ordinary  temperatures. 
Others,  such  as  the  metals,  are  apparently  quite  fixed,  but  they 
probably  possess  a  definite,  although  very  small  vapour-pressure, 
even  at  ordinary  temperatures.  Thus,  if  magnesium  is  heated  to 
550°  for  a  few  hours  in  a  magnesia  boat  enclosed  in  a  vacuous 
tube  it  sublimes  in  beautiful  crystals  on  the  cool  part  of  the  tube. 
The  vaporisation  of  a  solid  without  previous  fusion  is  called 
sublimation  ;  the  vapour -pressure  (like  the  vapour-pressure  of 
a  liquid),  is  definite  for  each  temperature,  is  independent  of 
the  volume  of  the  vapour  space,  and  increases  with  rise  of 
temperature. 

Eegnault  investigated  the  vapour-pressures  of  water  and  benzene,  both  in 
the  liquid  and  solid  states,  and  represented  his  results  graphically.  He  con- 
cluded that  the  curves  for  the  liquid  and  solid  joined  at  the  melting-point,  and 
gave  a  continuous  curve.  This  was  shown  theoretically  to  be  incorrect 
by  Kirchhoff  ( 1 858),  who  proved,  by  a  method  to  be  described  later,  that  the 


192  THERMODYNAMICS 

curves  must  at  least  have  different  tangents  at  the  melting-point.     Later 
experimenters  confirmed  this  conclusion. 

If  a  solid  is  heated  under  atmospheric 
pressure,  its  vapour-pressure  increases  with 
rise  of  temperature,  and  it  may  happen  that 
the  vapour-pressure  of  the  solid  becomes 
equal  to  atmospheric  pressure  before  the 
melting-point  is  reached.  In  this  case,  the 
substance  sublimes  away  without  previous 
fusion.  But  if  the  melting-point  is  reached 
before  the  vapour-pressure  reaches  atmo- 
spheric pressure,  the  substance  will  melt 
before  boiling  away. 
JIJG  37  The  two  cases  are  represented  in  Fig.  37. 

The  horizontal  isopiestic  cuts  the  vapour-pres- 
sure curve  of  the  solid  in  the  first  case,  that  of  the  liquid  in  the  second. 
Melting  can  be  brought  about  in  case  (1)  by  an  increased  pressure. 

The  vapour-pressures  of  ice  at  various  temperatures  have  recently  been 
carefully  determined;  Scheel  (1905)  finds  that  they  may  be  represented  by 
the  interpolation  formula  : 

logj>(mm.)  =  1 1-4796 -0-4  log  T-  26^'4. 
Kirchhoff's  formula  is  therefore  applicable  to  sublimation  (§  88,  Cor.  3). 

The  thermodynamic  treatment  of  sublimation  is  exactly 
analogous  to  that  of  evaporation.  Ramsay  and  Young  (1884) 
have  proved  experimentally  that  during  sublimation  the  tempera- 
ture remains  constant,  and  heat  is  absorbed ;  for  unit  mass  this 
is  the  latent-heat  of  sublimation,  L> 

We  have,  therefore,  the  corresponding  equations  : 

Evaporation.  Sublimation. 

(9.\         dpe L,  dps L^ 

dT  ~  T(A^)e  dT  ~  T(Arj, 


(3)  _ 

dT       RT2  dT       RT2 

(4)  log  j>.  =  A  -  ?  +  C  log  T        log  p.  =  A'  -  I'  +  C'  log  T. 

96.     Fusion. 

The  old  classification  of  bodies  into  solids,  liquids,  and  gases, 
based  on  differences  in  viscosity  and  elasticity,  is  not  altogether 
satisfactory.  We  shall  therefore  adopt  a  method  in  which  bodies 
are  divided  into  two  classes  according  to  the  nature  of  their 


CHANGES   OF   PHYSICAL   STATE  193 

internal  structure  (Lord  Kelvin,  Ency.  Britt.,  Art.  Elasticity, 
1878). 

Let  any  point  0  be  taken  in  the  interior  of  a  homogeneous 
body,  and  suppose  lines  OPi,  OPa,  OP,  .  .  .  drawn  in  different 
directions  through  0  (Fig.  38). 

If  now  the  physical  properties  of  the  body  (e.g.,  thermal 
expansion,  compressibility,  refractive  index,  electric  and  thermal 
conductivities,  dielectric  constant,  and  magnetic  permeability)  are 
measured  along  OPb  OP2,  OP,  ...  we  find  that  all  the  bodies  fall 
into  one  or  other  of  two  large  groups  :  — 

(1)  Bodies  in  which  the  physical  properties  are  identical  in 
all  directions  :  e.g.,  glass, 

air,  water.    This  class  of 

bodies  includes  all  gases, 

most    liquids,     and     the 

so  -  called      "  amorphous 

solids "    such    as    glasses 

(that    is,    solids    showing 

no     external     crystalline 

form,  and  breaking  with  a  glassy  fracture).      Bodies  of  this  type 

are  called  Isotropic  Bodies. 

(2)  Bodies  in  which  some,  or  all,  of  the  physical  properties 
are  different  in  different  directions.      This  class  includes  crystal- 
line solids,  and  liquid  crystals.      Thus  all  crystals  except  those 
belonging  to  the  regular  system  are   optically  different   along 
different  axes;  crystals  of  the  regular  system,  although  optically 
isotropic,  show  differences  in  electrical  properties  along  different 
directions.      Bodies  of  this   type,  the  properties  of  which  are 
vectorially  distributed  in  space  are  called  Anisotropic  Bodies  (or, 
JEolotropic  Bodies). 

Between  isotropic  and  anisotropic  solids  there  is  another 
marked  distinction,  based  on  their  behaviour  when  heated. 
Whereas  the  former  (e.g.,  glass)  gradually  soften,  and  pass  con- 
tinuously, through  various  grades  of  plasticity,  into  more  or  less 
mobile  liquids,  the  anisotropic  bodies  exhibit  a  sharply  denned 
melting-point,  i.e.,  a  definite  temperature  at  which  transition  from 
solid  to  liquid  occurs.  The  process  of  softening  throughout  the 
mass,  characteristic  of  isotropic  bodies,  is  absent,  and  fusion 
occurs  only  at  the  sharply  defined  boundary  between  the  crystal 
and  its  melt.  (Softening  before  fusion,  in  which  distortion,  but 

T.  o 


194 


THERMODYNAMICS 


not  destruction,  of  the  crystals  occurs,  is  exhibited  by  a  few 
crystalline  solids,  such  as  iron,  platinum,  and  sodium;  it  is 
applied  in  the  process  of  welding.) 

Tammaun  has  advanced  the  view  that  "amorphous  solids'"  are  really 
liquids  which  have  been  cooled  far  below  their  freezing-points,  and  have 
thereby  acquired  great  viscosity,  but  have  not  crystallised.  They  are  super- 
cooled liquids.  This  hypothesis  is  supported  by  the  following  evidence  : 

(1)  Such  bodies  have  no  definite  melting-point,  but  pass  continuously  into 
the  liquid  state  on  heating. 

(2)  They  show  signs  of  plasticity  when  submitted  to  prolonged  stresses 
(pitch,  glass,  sealing-wax,  etc.,  "flow"  very  slowly). 

(3)  A  liquid  may  often  be  rapidly  supercooled  into  a  glassy  condition. 

(4)  The  amorphous  state  frequently  passes  spontaneously  into  the  crystal- 
line state  (plastic  sulphur,    "devitrification"'  of  glass,  Gore's  amorphous 
antimony). 

Although  Carnelley  once  thought  he  had  been  able  to  super- 
heat ice  ("hot  ice"),  it  is  almost 
certain  that  no  solid  can  be  main- 
tained alone  at  a  temperature  higher 
than  its  melting-point.  Tammann 
(Zeitschr.  plujsik.  Chem.,  68,  257, 
1910)  finds,  however,  that  a  crystal- 
line solid  may,  under  certain  cir- 
cumstances, be  superheated  in  the 
presence  of  its  melt.  This  occurs 
when  the  supply  of  heat  to  the 
crystal  is  sufficiently  great  in  com- 
parison with  the  linear  velocity  of 
-  crystallisation  of  the  supercooled 
liquid  (cf.  Findlay :  Phase  Ride). 
The  impossibility  of  realising  a 
superheated  crystal  is  ascribed  by  Tammann  to  the  very  large 
number  of '"  centres  "  in  the  crystal  in  which  fusion  can  commence, 
in  contrast  to  the  relatively  small  number  of  isolated  points 
in  the  supercooled  liquid  where  crystal-clusters  begin  to  form. 

The  changes  of  volume,  and  the  quantities  of  heat  absorbed 
during  fusion,  are  much  less  than  the  changes  which  accompany 
evaporation  : 

Substance       At-  (fusion)  Av  (evap.)  Lf        Le 

Water  —  0'125  c.c.         +  1646  c.c.         80  cal.  536  cal. 

Acetic  Acid      +  0'121  c.c.         +    385  c.c.         43  cal.    97  cal. 


6m, 


FIG.  39. 


CHANGES   OF   PHYSICAL   STATE  195 

The  changes  of  volume  accompanying  fusion  were  first  noticed 
by  Reaumur  (1726),  who  observed  that  when  a  fused  mass  solidi- 
fied in  a  crucible,  the  surface  was  usually  concave,  indicating 
contraction  on  solidification,  but  in  a  few  cases  was  convex, 
indicating  expansion.  Wax  and  water  being  common  examples 
of  the  two  classes,  we  may  speak  of  bodies  of  the  wax-type  or  ice- 
type,  respectively,  according  as  they  expand  or  contract  on  fusion. 
The  curves  for  r  =  /  (9)  in  each  case  exhibit  a  discontinuity  at  the 
melting-point  ;  in  the  first  case  the  curve  rises,  in  the  second  it 
falls  *  (Fig.  39).  The  volume-changes  accompanying  fusion  have 
been  measured  by  the  dilatometeu  (Pettersson,  1881,  etc.).  The 
majority  of  substances  belong  to  the  wax-type  ;  water,  bismuth, 
bismuth  sulphide,  cast-iron,  nitre,  and  some  alloys,  belong  to  the 
ice-type. 

97.     Thermodynamics   of   Fusion. 

The  effect  of  change  of  pressure  on  the  melting-point  of  a 
substance  can  be  predicted  qualitatively  in  the  following  manner  : 

Let  a  Carnot's  cycle  be  carried  out  between  the  temperatures 
T  and  (T  —  ST)  with  a  mixture  of  solid  and  liquid  as  the  work- 
ing substance.  During  the  fusion  process  heat  is  invariably 
absorbed,  hence  fusion  must  occur  along  the  isotherm  correspond- 
ing to  the  temperature  of  the  source.  Now,  there  are  two  cases  : 
(i.)  The  substance  expands  on  fusion  (wax-type) — the  T  isotherm 
is  therefore  traced  out  from  left  to  right  on  the  indicator  diagram, 
and  since  the  (T  —  ST)  isotherm  is  necessarily  traced  in  the 
opposite  direction,  and  the  cyclic  area  is  positive  and  therefore 
traced  out  clockwise  (by  Carnot's  theorem),  it  follows  that  the 
(T  —  8T)  isotherm  lies  below  the  T  isotherm.  (ii.)  The 
substance  contracts  on  fusion  (ice-type) — the  T  isotherm  is  traced 
out  from  right  to  left,  and  the  above  reasoning  shows  that,  in 
this  case,  the  (T  —  8T)  isotherm  lies  above  the  T  isotherm. 

Along  each  of  these  isotherms  solid  and  liquid  are  in  equili- 
brium ;  each  corresponds  to  a  melting-point  under  a  given  pres- 
sure. Thus  we  see  (qualitatively)  that  the  melting-point  of  a  sub- 
stance of  the  wax-type  is  raised  by  increasing  the  pressure  ;  that  of 
a  substance  of  the  ice-type  is,  on  the  other  hand,  lowered. 

*  According  to  Kopp,  the  curve  is  really  continuous ;  Pettersson  and 
later  workers  consider  that  this  is  the  case  only  in  presence  of  impurities. 

o  2 


196 


THERMODYNAMICS 


This  conclusion  was  arrived  at,  from  considerations  based  on 
Carnot's  principle  alone,  by  James  Thomson  in  1849.  He  also 
calculated  the  magnitude  of  the  effect,  in  the  case  of  ice,  by 
means  of  a  cyclic  process.  Since  the  reasoning  is  the  same  for 
both  cases,  we  shall  deal  with  both  together,  giving  appropriate 
diagrams. 

A  mixture  of  solid  and  liquid  in  equilibrium  at  a  temperature 
(T  —  8T)  under  a  pressure  (p  —  fy>),  is  taken  round  a  small 
Carnot's  cycle  ABCD. 

Work  done  in  cycle  =  FB  .  BC 


where  the  symbols  have  the  same  significance  as  in  §  88,  except 


p 

(Wax  Type) 

Q                 \B             \C 

5   T 

P 

V 

V 

R  (T-6T) 

F 

\       E 

\ 

X  Y 

FIG.  40. 


P 

(Ice   Type) 

P         0\ 

E      A\F              /?  

,       \ 

\ 

s  T 

c 

^           B 

x 
FIG.  41. 


that    n,    i-2,    are     now    the    specific    volumes    of     solid    and 
liquid. 

Heat  absorbed  from  source  =  niLf, 

.'.    JF  8T  X  Ar  X  m  =  mLf  X  -^,  by  Carnot's  theorem, 


_ 

•  dp  ~    L, 

Thus,  for   small   changes,   the   change   of  melting-point,   or 
freezing-point,  with  pressure  is  given  by  : 

.         .'      .         .     (2) 

There  are  two  cases  possible  : 

(i.)  Ar  is  >  0  (wax-type)  .'.  8T  has  the  same  sign  as  bp,  or 
the  melting-point  is  raised  by  increase  of  pressure  (Fig.  40). 


5T  =  ~  bp 


CHANGES  OF   PHYSICAL   STATE  197 

(ii.)  Ar  is  <  0  (ice-type)  .*.  8T  and  bp  have  opposite  signs,  or 
the  melting-point  is  lowered  by  increase  of  pressure  (Fig.  41). 

In  the  case  of  water  : 

L/  =  80-4    gr.    cal.    per   gram  =  4-2  x  107  X  80-4    ergs,    v*  =  1-000   c.c., 
vl  =  1-091  c.c.,  T  =  273°. 

If  Sp  =  I  atm.  =  1013130  dyne/cm.a, 


The  prediction  of  James  Thomson  was  verified  experimentally  by  his 
brother,  Lord  Kelvin,  in  1850,  who  found  ST  =  —  0'0072°  C.  per  atm. 
Dewar  (1880)  found  that  this  remained  practically  constant  up  to  700  atm. 

Bunsen  (1850),  Hopkins  (1854),  Batelli  (1887),  and  de  Vissier  (1892)  also 
made  experiments  on  the  effect  of  pressure  on  the  melting-point  of  bodies 
of  the  wax-type.     The  latter  found  for  acetic  acid  : 
5T  =  +  0-02435°  C.  per  atm.  (obs.). 

Now  L,-  =  46-42  g.  cal.,  T  =  289-6°,  A  v  =  +  0-0001595  litre 
.  • .  *T  =  +  0-0242JC.  per  atm.  (oalc.). 

The  data  in  this  field  have  recently  been  greatly  extended,  especially  by 
Tammaun,  to  whose  monograph :  Kristallisieren  und  ticltmehen,  Leipzig, 
1903,  the  reader  is  referred  for  a  detailed  account  of  the  subject. 

The  equation  of  Planck  (§  93)  for  the  dependence  of  the  latent 
heat  of  a  change  of  state  on  the  temperature  and  pressure  applies, 
of  course,  to  fusion  as  well  as  evaporation  : 

T  /T  T  f   /^   rf\  /^'\      \ 

In  the  case  of  ice  at  0°  C. : 
<•/'  =  1-01  (water)  r"  =  TOO  ~  =  —  O'OOOOG 

cj  =  0-50  (ice)  <•'  =  1-09  ^  =  O'OOOl 

L,  =  80  T  =  273 

.-.  ^'  =  0-66 

i.e.,  if  the  melting-point  of  ice  is  lowered  1°  C.  by  increase  of 
external  pressure,  the  latent  heat  of  fusion  decreases  by  0'66  cal. 
The  approximate  equation  : 

dT  =  c""  ~  c"' 

was  used  by  Person  so  early  as  1847.  Si  nee  in  nearly  all  cases 
Cp"  >  Cp,  the  latent  heat  of  fusion  increases  with  rise  of  tempera- 
ture, in  contrast  with  the  latent  heat  of  evaporation. 


198 


THERMODYNAMICS 


98.  Allotropic    and   Polymorphic   Change. 

If  two  or  more  crystalline  forms  of  a  substance  exist,  they  are 
called  polymorphic  forms ;  if  the  substance  is  an  element  (C,  S,  P) 
they  are  called  allotropic  forms.  The  Thomson  equation  obviously 
applies  to  this  case  : 

rjT_  T(ra  -  n) 

dp  ~        LB 

where  T  =  transition  temperature ;  LM  =  latent  heat  of  the 
transition ;  TI,  v%  =  specific  volumes  of  the  denser  and  lighter 
forms,  respectively. 

Keicher  (1883)  applied  this  to  the  sulphur  transition  : 

Sa— >  SjS,  in  which 
»;2  -  ri  =  0-0000126  c.c. ;  LH  =  2'52  cal.  ;  T  =  273  +  95°-6, 

7m 

.-.  °J    =  +  0-045.     (Obs.  0-05°  C.  per  atm.) 

Polymorphic  transition  may  occur  very  rapidly  (e.g.,  tetrabroin  methane, 
boracite),  but  usually  takes  a  fairly  long  time  (e.g.,  sulphur).  In  some  cases 
there  is  no  apparent  change  in  finite  time  (e.g.,  diamond  to  graphite).  In  all 
cases  a  rise  of  temperature,  and  contact  with  the  second  form,  accelerate  the 
process. 

99.  False  Equilibrium. 

If  the  pressure  or  temperature  of  a  system  of  two  phases,  a  and 
/3,  in  true  equilibrium,  is  altered,  even  infinitesimally,  the  one 
phase  passes  over  completely  into 

the  other,  the  change  a >  ft,  or 

the  change  ft >  a,  taking  place 

according  as  the  temperature,  or 
pressure,  is  greater  or  less  than  the 
value  corresponding  with  equi- 
librium. The  rapidity  of  the 
change  is,  when  the  pressure  or 
temperature  difference  is  not  too 
large,  regulated  by  the  rate  at 
which  heat  is  supplied  to,  or  with- 
drawn from,  the  system.  If, 

however,  experiments  are  made  over  an  equilibrium  curve 
extending  through  a  wide  range  of  temperature,  it  is  found  that 
the  velocities  of  transition  at  low  temperatures  are  very  much  less 
than  would  be  expected  from  the  diminished  heat-flow  alone.  At 


p  kg /cm* 


FIG.  42. 


CHANGES   OF   PHYSICAL   STATE  199 

some  low  temperature  the  transition  may  cease  altogether,  and 
the  two  forms  may  coexist  under  conditions  such  that  one  form 
should  pass  completely  over  into  the  other. 

A  very  instructive  case  is  the  transition  of  the  two  crystalline  varieties  of 
phenol,  phenol  I  and  phenol  II,  studied  by  Tainmann  (Fig.  42).  Above  25°  the 
transition  occurs  reversibly,  phenol  I  passing  into  phenol  II  with  evolution 
of  heat  and  diminution  of  volume.  If,  at  a  given  temperature,  greater  than 
25°,  the  pressure  is  changed  from  the  equilibrium  value  to  a  value  greater 
or  less  than  this,  the  form  I  passes  into  form  II,  or  vice  versa,  until,  after 
10—20  minutes,  the  equilibrium  pressure  is  recovered.  Below  25=  the 
relations  are  quite  different.  Thus  at  20°,  a  difference  of  28  kg.  cm.'2 
remains  between  a  rising  and  a  falling  pressure  after  half  an  hour,  and  the 
difference  increases  with  fall  of  temperature  until  at  —  21°  it  is  no  less  than 
600  kg.  cm.2.  The  curve  of  transition  down  to  25°  is  one  of  true,  or  rever- 
sible equilibrium  ;  at  lower  temperatures  it  is  divided  into  two  parts,  called 
by  Tammanu  "  limitative  curves,"  AC  and  BC,  enclosing  a  region  in  which 
both  phases  coexist  in  intimate  contact,  if  not  indefinitely  at  least  for  a  con- 
siderable period  of  time.  This  region  is  called  by  Taninianu  the  region  of 
pseudo-equilibrium,  and  by  Duhem  the  region  of  false  equilibrium.  If  the 
pressure  is  changed  so  as  to  pass  outside  the  region  ABC,  it  slowly  comes 
back  to  the  value  on  one  of  the  limiting  curves,  and  then  ceases  to  change. 
There  are  two  possible  interpretations : 

(i.)  The  velocity  of  transition,  which  is  known  to  decrease  very  rapidly 
with  decrease  of  temperature,  has  become  so  small  that  no  appreciable  change 
occurs  in  ordinary  periods  of  time. 

(ii.)  The  transition  has  ceased  altogether,  and  the  system  is  in  a  state  of 
false  equilibrium.  Two  types  are  recognised  (Duhem,  Traite  tie  Mecanique 
chimique,  I,  ii) : 

(«)  Those  in  which  two  or  more  forms  coexist  under  conditions  where  only 
one  form  is  theoretically  stable — "  faux  equilibres  reels  "  (Duhem) ;  "  Psuedo- 
gleichgewichte  "  (Tammann).  E.g.,  the  two  forms  of  phenol. 

(b)  Those  in  which  one  form  is  existing  in  a  metastable  state  under  con- 
ditions whei'e  the  other  form  is  stable,  and  destroyed  by  the  introduction  of 
a  trace  of  the  second  form — "faux  equilibres  apparent"  (Duhem)— meta- 
stable equilibria.  E.g.,  superheated  or  supercooled  liquids.  The  existence  of 
such  states  is  closely  connected  with  the  phenomena  of  capillarity  (Duhem,  loc. 
cit.  II.,  2,  also  pp.  66  et  seq. ;  III.,  121.  Gibbs,  Xcientif.  Papers,  I.  252  et  seq.}. 
All  cases  are  covered  by  a  very  general  theorem  to  the  effect  that  whereas, 
when  the  thermodynaniic  conditions  of  equilibrium  are  satisfied  the  system 
will  be  in  equilibrium,  the  converse  is  not  always  true  (Moutier,  1880). 

100.     Influence  of  Compression. 

In  the  preceding  considerations  of  vapour-pressure  it  has  been  assumed 
that  the  pressure  is  uniform  throughout  the  whole  system.  A  condensed 
phase  may,  however,  exist  under  a  pressure  different  from  that  of  its  accom- 
panying vapour,  as  in  the  following  cases ; 


200  THERMODYNAMICS 

(i.)  A  liquid  or  solid  phase  may  exist  under  a  pressure  greater  than  its 
vapour-pressure  if : 

(a)  An  indifferent  gas  is  pumped  into  a  closed  vessel  containing  the  two 
phases. 

(#)  Liquid  is  put  in  contact  with  vapour  through  a  capillary  tube,  or 
collection  of  such  tubes  ("  sieve  "),  not  wetted  by  the  liquid,  and  is  compressed 
from  behind. 

(y)  A  solid  is  held  in  a  net  of  wire  gauze  which  exerts  compression  on  it. 

(ii.)  A  liquid  may  exist  under  a  pressure  less  than  its  vapour-pressure 
when  it  wets  and  rises  in  a  capillary  tube  placed  in  the  liquid,  and  exposed 
above  to  the  pressure  of  the  vapour  alone. 


Let  p  be  the  ordinary  equilibrium  pressure  in  a  system  com  • 
posed  of  a  condensed  phase  and  its  vapour,  and  let,  r,  V  be  the 
specific  volumes  of  these  phases,  respectively,  under  a  pressure  p. 
We  now  assume  that  these  values  are  altered  to  p',  r',  V,  when 
the  condensed  phase  alone  is  exposed  to  a  pressure  P  +  p. 

Now  r'  =  r  —  r>jP,  where  rj  is  the  coefficient  of  compressi- 
bility (§  23),  since  for  all  pressures  the  relative  diminution  of 
volume  of  a  fluid  is  proportional  to  the  increase  of  pressure. 

We  assume  that  the  following  isothermal  reversible  cycle  may 
be  carried  out : 

(i.)  Evaporate  unit  mass  of  the  condensed  phase  at  the  pressure 
P  +  p,  so  that  the  vapour  produced  is  at  the  pressure  p', 
the  work  done  =  p'V  —  (j)  -\-  P)  r  (1  —  Pr?r). 

(ii.)  Change  p' ,  V  to  p,  V  by  expansion, 

f" 
the  work  done  =        pdv 


=J 


P' 
if  the  vapour  obeys  Boyle's  law. 

(iii.)  Condense  the  vapour  at  the  pressure  p, 

the  work  done  =  —  |>(V  —  r). 
(iv.)  Compress  the  condensed  phase  to  pressure  P  +  p, 

I         P\ 
the  work  done  =  —  (p  +  -5  j  P?jr. 

The  cycle  is  now  completed,  hence,  by  §  36  : 


CHANGES   OF   PHYSICAL    STATE  201 

or,  if  we  put  pV  =  p'\'  (Boyle's  law),  and  neglect  terms  con- 
taining rip, 


.'.  if  P  is  not  very  great, 


p 
p~ 

=  1  +  ~  +  etc., 


"       *  ~       v 

Hence  if  6P  is  the  increased  pressure  applied  to  the  condensed 
phase,  and  ftp  the  consequent  increase  of  vapour-pressure, 

. 

This  equation  is  due  to  Willard  Gibbs  (1876). 

The  vapour-pressure  of  any  liquid  or  solid  is  increased  by 
compression  of  the  condensed  phase. 

If  m  is  the  molecular  weight,  and  if  the  vapour  obeys  the  gas 
laws, 

?=ik'p v 

where  $  is  the  molecular  volume  of  the  condensed  phase.  Thus 
the  relative  increase  of  vapour-pressure  for  a  given  increase 
of  pressure  on  the  condensed  phase  depends  solely  upon,  and  is 
proportional  to,  the  molecular  volume  of  the  latter,  at  constant 
temperature. 

Example. — The  specific  volume  of  ice  at  0°  is  1-092 

...  $P  =    *   ^_1^92xlO-«/.     per  atm 
±U        0.082  /.  atm.  x  273 
deg. 

.-.  rise  of  vap. -press.  =  0-088  per  cent,  per  atm. 

The  vapour-pressure  of  a  liquid  contained  in  a  long  vertical  column  exposed 
to  the  action  of  gravity  is  greater  in  the  lower  part,  and  if  p0,pi  are  the 
vapour-pressures  at  points  distant  /;0,/h  from  a  fixed  horizontal  plane  of 
reference  (J>i  >  J>0) : 


where  </  is  the  intensity  of  gravity  (W.  Gibbs). 


202  THERMODYNAMICS 

101.     Effect  of  Surface  Tension. 

The  tension  existing  in  the  superficial  layer  of  a  liquid  tends  to 
make  any  mass  of  the  latter  take  up  the  shape  having  the  mini- 
mum area  of  exposed  surface  for  a  given  volume,  and  in  the  case 
where  gravity  is  inoperative,  or  negligible,  a  drop  of  liquid  is  always 
a  well-defined  sphere.  The  surface  forces  give  rise  to  a  mechanical 
pressure  P,  exerted  along  the  inward-drawn  normal,  the  value  of 
which  may  be  calculated  from  the  principle  that  the  free  energy 
of  the  system  in  equilibrium  is  a  minimum,  or  : 
3*  =  -  5A  =  0. 

Consider  a  spherical  drop  of  radius  r,  and  suppose  this  under- 
goes a  virtual  displacement  such  that  r  is  changed  to  r  -j-  Sr. 
The  work  done  in  the  displacement  vanishes  to  the  first  order. 

The  increase  of  surface  =  Simlr  —  da 
and  increase  of  volume  =  4in*dr  =  dr, 
/.-5A=  -  Prfy  +  rrda  =  Q 

where  a-  =  surface  tension, 
=  0, 


.-.p=|.     .....  CD 

If  the  surface  is  plane,  r  =  oo  /.  P  =  0.  It  is  probable  that  a 
pressure  K  is  exerted  on  the  liquid  by  a  plane  surface,  due  to 
molecular  attraction,  so  that  equation  (1)  should  be  written 

P  =  K  +  ~    .....     (1«) 

The  magnitude  K  cannot,  however,  be  determined  by  measure- 
ments of  capillary  magnitudes  alone,  and  we  shall  not  consider  it 
any  further,  understanding  by  P  the  increase  of  pressure,  over 
that  exercised  by  a  plane  surface,  due  to  curvature. 

It  can  be  shown  generally  that,  if  the  surface  has  two  principal 
radii  of  curvature  i\,  r2,  then  : 


If  the  surface  is  spherical,  i\  —  r2  =  r  .'.  P  =  <r/2r. 

Example.  —  Find  the  excess  of  pressure  inside  a  soap-bubble  1  mm.  in 
diameter  over  the  atmospheric  pressure.  (Surface  tension  of  water  =  81  ergs 
per  sq.  cm.)  [1(5  X  10~5  atm.] 

It  is  an  immediate  consequence  of  the  result  of  the  preceding 
paragraph  that  the  vapour-pressure  of  a  liquid  in  the  form  of  drops 


CHANGES  OF  PHYSICAL   STATE  203 

will  be  greater  than  that  of  a  mass  of  liquid  having  a  plane 
surface,  by  reason  of  the  hydrostatic  pressure  exerted  on  the 
liquid  in  the  drop  by  capillary  forces. 

The  influence  will  be  greater  the  less  the  radius  of  the  drop. 

For/>-V  =  ^P  =  £.^     ...     (3) 

in  the  case  of  a  sphere ;  or 


in  the  case  of  a  surface  of  any  form. 

This  equation  is  due  to  Lord  Kelvin  (1870). 

If  the  curved  surface  is  convex,  as  in  the  case  of  liquid  drops 

or  the  surface  of  mercury  depressed  in  a  capillary        ^ ^ 

tube,  p'>p,  but  if  it  is  concave,  as  in  the  case  of  a 
liquid  ascending  and  wetting  a  capillary  tube,  r  is 
negative  and  p'<p.  The  application  to  suspended 
ebullition  (§  88)  is  quite  obvious. 

The  capillary  tube  cases  are  readily  proved  directly,  for  it 
is  evident  from  Fig.  43  that  the  pressure  just  beneath  the 
concave  surface  is  less  than  that  just  beneath  the  plane 
surface  by  an  amount  hfi;  whereas  the  vapour-pressures 
just  above  these  surfaces  differ  by  Jtj\.  Hence 
p—p'  =  h!Vi  P  =  *•'?, 

.'.    p  —  p'  =  P  !L  as  before. 

Cantor  (1895)  has  indicated  that  the  deic-point,  or  temperature 
at  which  vapour  condenses  on  a  solid  surface,  must  be  different 
from  the  saturation  temperature  for  the  vapour  over  the  surface 
of  its  own  liquid,  because  of  the  different  surface  tensions  between 
(vapour)/(solid)  and  (vapour  )/(liquid). 

The  influence  of  surface  energy  on  the  melting-point  of  a  solid 
has  been  calculated  by  P.  Pawlow,  who  found  experimentally  that 
with  salol  granules  of  surfaces  228 — 1296  /u-,  the  m.pt.  decreased 
2-8°  C.  per  100  times  increase  of  surface  (p.  =  O'OOl  mm.). 

W.  Thomson,  Phil.  Mag.  [4],  4?,  448,  1871 ;  cf.  E.  Warburg.  Wied.  Ann. 
28,  394,  1886;  B.  v.  Helmholtz,  ibid.  27,  522,  1886;  G.  F.  Fitzgerald,  PhiJ. 
Mag.  [5],  8,  382,  1879;  J.  Stefan,  Wied.  Ann.  29,  655,  1886;  B.  Galitzine, 
ibid.  35,  200,  1888;  G.  F.  Fitzgerald,  Mature,  49,  316,  1894;  A.  Boek,  BeiU. 
20,  361,  1896;  A.  Bacon,  Phys.  Rev.  20,  1,  1905. 

M.  Cantor,  Weid.  Ann.  56,  492,  1895. 

P.  Pawlow, Vonrn.  Buss.  Chem.  Soc.  40,  1052,  1910. 


204  THEKMODYNAMICS 

102.     Influence  of  Electrification  of  the  Surface  of  a  Phase. 

If  a  spherical  drop  of  water  in  the  air  carries  an  electrostatic  charge,  as  is 
the  case  with  the  drops  composing  a  thunder-cloud,  this  charge  resides  in 
the  surface,  and  it  is  readily  shown  (cf.  J.  J.  Thomson,  Elements  of 
Electricity  and  Magnetism,  §  37)  that  a  charge  of  surface  density  k  causes 
a  hydrostatic  tension  2ir&2  to"  be  exerted  on  every  unit  of  surface,  acting  along 
the  outward-drawn  normal,  and  so  tending  to  explode  the  drop. 

By  equation  (1)  of  §  100  this  will  cause  a  reduction  of  vapour-pressure  to 
the  extent  : 

1>_J/=_8p  =  p!L=_2,A^    .        .         .       (1) 

an  equation  deduced  by  Blondlot  (Journ.  de  Phys.  [2],  3,  442,  1884). 

The  effect  is,  however,  small,  because  it  is  known  that  the  greatest  tension 
which  can  exist  on  an  isolated  conductor  in  air  under  atmospheric  pres- 
sure is  equal  to  a  pressure  of  about  0'3  mm.  of  mercury  ;  this  corresponds 
to  a  lowering  of  vapour-pressure,  in  the  case  of  water,  of  only  about  10  ~  6  mm. 
of  mercury. 

Gouy  (C.  R.  149,  822,  1909)  has  shown  that  Blondlot's  equation  is  incom- 
plete ;  the  correct  equation  is  : 


where  K  is  the  dielectric  constant  of  the  vapour. 

The  effect  of  a  magnetic  field  has  been  considered  by  Duhein  (1890),  and 
Koenigsberger  (Ann.  Phijs.  66,  709,  1898). 

103.     Supposed    Critical    Point     of    Fusion  ;    Researches    of 
Tammann. 

Poynting  (1881),  from  considerations  based  on  equation  (1)  of 
§  100,  surmised  that  ice  would  melt,  if  the  pressure  on  it  is  raised 
in  such  a  way  that  any  water  produced  flows  away,  under  a 
pressure  which  is  for  a  given  temperature,  only  ^  that  on  the 
fusion  curve  (i.e.,  when  the  water  remains  in  contact  with  the 

ice).     He  found  -r-  /-JT;  =  —  -  -  =  O'l,  i.e..  the  ordinary  fusion 
dpi  dP  n 

curve  rises  ten  times  as  fast  as  the  so-called  "  second  fusion  curve." 
Tammann  pointed  out  that  the  validity  of  the  calculation  stands 
or  falls  with  the  correctness  of  the  assumption  that  fusion  actually 
occurs  in  process  (i.)  of  the  cycle.  He  compressed  ice  under  a 
loose  piston,  and  found  that  although  it  certainly  became  very 
plastic  near  the  melting-point,  the  velocity  of  outflow  increased 
continuously  from  low  temperatures,  thus  excluding  the  possibility 
of  a  second  fusion  curve  having  been  intersected. 

Poynting  also  expressed  the  opinion  that  a  critical  point  of 


CHANGES   OF  PHYSICAL   STATE 


205 


fusion,  at  which  solid  would  pass  continuously  throughout  its 
mass  into  liquid,  might  exist  on  the  fusion  curve.  The  horizontal 
part  of  the  curve  of  fusion  changes  in  length  with  change  of 
temperature,  and  at  some  high  or  low  temperature  (according  as 
the  substance  belongs  to  the  wax  or  ice-type)  might  shrink  to  a 
critical  point. 

Tainrnann  has,  by  a  large  amount  of  experimental  evidence, 
apparently  refuted  the  hypothesis  of  a  critical  point  of  fusion — 
against  which  of  course  there  is  no  a  priori  objection.  In  some 
cases  (e.f/.,  with  ice)  new  crystalline  modifications  appear  when 
the  temperature  and  pressure  are  modified,  and  in  all  cases 
the  pair  of  relations  :  L  =  0,  Ar  =  0  (cf.  §  89)  were  never 
simultaneously  satisfied. 

The  general  form  of  diagram  for  the  transition  : 

Crystalline  Solid >  Liquid, 

where  the  specific  volumes  of  solid  and  liquid,  and  the  latent 
heat  are  represented  as  functions  of  the  equilibrium  pressure 
(and  therefore,  implicitly,  of  the  tem- 
perature) is  shown  in  Fig.  44.  It 
differs  completely  from  the  correspond- 
ing diagram  for  evaporation  (cf. 
Fig.  30 ;  the  L(  curve  slopes  down  to 
meet  the  p  axis  at  the  abscissa  of  K). 
The  specific  volume  curves  intersect 
at  a  point  where  Ar  =  0;  the  latent 
heat,  on  the  contrary,  changes  only 
very  slightly  with  the  temperature, 
and  its  curve  is  either  horizontal, 
or  exhibits  a  maximum,  falling  off  slightly  at  higher 
pressures,  and  probably  approaching  the  p  axis.  Thus,  when 
Ar  =  0,  L,  has  a  considerable  positive  value,  and  when  L,-  —  0, 
Ar  has  (probably)  a  considerable  negative  value. 

The  melting-point  (T,j9)  curve  (unlike  a  vapour-pressure  curve  of 
a  liquid)  does  not  end  abruptly  at  a  critical  point  (Ar  =  0,  L  =  0) ; 
it  is  an  endless  curve,  probably  forming  a  closed  loop  ABCD, 
unless  it  intersects  some  other  curve  or  the  axes  of  co-ordinates. 
At  high  pressures  it  bends  round  towards  the;;  axis,  and  according 
to  Tammann,  takes  the  shape  indicated  by  the  following  con- 
siderations. It  is  known  from  experiment  that  (for  substances 
of  the  wax-type)  the  melting-point  increases  with  rise  of  pressure, 


FIG.  44. 


206 


THEEMODYNAMICS 


but  more  slowly  as  the  pressure  increases, 
by  the  numbers  for  naphthylamine  : 


This  is  well  shown 


p  atm. 

r»p. 

1 

49-75 

62 

50-49 

81 

50-54 

93 

50-33 

143 

50-01 

166 

49-83 

173 

49-65 

After  a  certain  point  the  melting-point  begins  to  fall 
hence  it  must  pass  through  a  maximum  (Fig.  45). 


again, 


AtB     ,     T-  =  0    ,     /.    Ar  =  0; 


atC     ,    £  = 


Liq. 


L,  =  0. 


Crystal        C\   Gtess 


Inside  ABCD,  the  crystalline  solid  is 
stable,  above  ABC  the  liquid,  whilst  to 
the  right  of  CD  the  stable  form  is  the 
amorphous  glass.  Roozeboom,  however, 
holds  a  different  opinion  as  to  the  latter 
part  of  the  curve  (Heterogcne  Gleichge- 
tcichtc,  vol.  I.). 

The  dependence  of  melting-point  on  pressure  was  found  to  be 
well  represented,  up  to  several  thousand  atmospheres,  by  the 
equation  proposed  by  Daniien  (1891)  : 

0lt  =  0p=l  +  a  (p  -  1)  -  6  0>  -  I)2- 
Thus  p  is  a  maximum  for 


FIG.  45. 


and  pnnix*  6,IMX  can  be  calculated  from  the  empirical  constants 
a  and  b. 

In  some  cases  the  constants  hold  good  up  to  a  certain  point 
only,  when  the  curve  suddenly  changes  its  direction.  This  is 
attributed  to  the  appearance  of  a  new  crystalline  solid  phase,  and 
three  varieties  of  ice  have  been  so  discovered. 

On  the  right  of  the  curve,  in  the  region  marked  "  glass,"  we 


CHANGES  OF  PHYSICAL   STATE  207 

may  have  states  of  false,  or  fixed,  equilibrium.  To  the  right  of 
the  dotted  line  CE  the  substance  is  in  the  amorphous  state,  and 
if  we  subject  crystals  to  pressures  greater  than  the  abscissae  of 
this  line,  they  pass  into  the  amorphous  state,  whilst  at  pressures 
less  than  the  abscissae  of  CD,  the  amorphous  substance  crystal- 
lises. In  the  region  DCE,  both  crystalline  and  amorphous  states 
co-exist  in  a  condition  of  false  equilibrium. 

In  conclusion,  we  may  observe  that  the  phrase  "  solid  state  ''  is  indefinite, 
a  better  classification  of  states  is  the  following  (Tammann) : 


A.  Isotropic  States. 

(1)  Gaseous. 

(2)  Liquid. 

(3)  Amorphous  solid, 

or  Glass. 


B.  Anisotropic  States. 

(1)  Crystalline  solid  ;    different 

polymorphic    crystalline 
varieties. 

(2)  Crystalline  liquid. 


Continuous  transition  of  state  is  possible  only  between  isotropic  states  ;  it 
may  thus  occur  between  amorphous  glass  (i.e.,  supercooled  liquid  of  great 
viscosity)  and  liquid  ("  sealing-wax  type  of  fusion  "),  or  between  liquid  and 
vapour,  but  probably  never  between  anisotropic  forms,  or  between  these 
and  isotropic  states.  This  conclusion,  derived  from  purely  therniodynaniic 
considerations,  is  also  supported  by  molecular  theory. 

104.     Dissociation. 

Although  Deville  and  Debray  had  established  the  main  laws 
of  the  dissociation  of  substances  by  heat,  it  was  reserved  for 
A.  Horstmann  to  show  that  the  dissociation  pressure  of  sal- 
ammoniac,  for  example  : 

NHjCl  7—*  NH3  +  HC1 

solid 

gas 

is  dependent  on  the  temperature  in  the  same  way  as  the  vapour- 
pressure  of  a  liquid,  and  hence  the  latent-heat  equation  (§  88) 
can  also  be  applied  to  chemical  phenomena. 

The  dissociation  pressure  is  a  function  of  temperature  alone  : 

P  =  <KT)      .....     (1) 

If  VxHjCi,  VHC1,  YxH3  are  the  molecular  volumes  of  the  solid 
and  two  gases,  respectively, 


AY  =  VXH,  +  YHC1  -  VMI.CI  =  2V  =  .         .         .     (2) 

if  the  volume  of  the  solid  is  negligible  compared  with  that  of  the 
gas,  and  the  latter  obeys  the  gas  laws. 


208  THERMODYNAMICS 

The  external  work  is  therefore 

A  =  p&V  =  2  RT  .  .  .  .  (3) 
and  if  Arf  is  the  heat  absorbed  in  the  process  —  the  heat  of 
dissociation,  we  have  (§  58)  : 


If  n  mols  of  gas  are  produced  instead  of  2  : 


The  unit   of  pressure  has   no   influence;  if  Ad  is  in  gr.  cal., 
R  =  1'985.     In  the  calculation  of  -j/L  we  may  use  any  one  of  the 

three  methods  : 

(i.)  Graphically,  by  drawing  a  tangent  to  the  p,T  curve. 
(ii.)  Algebraically,  by  differentiating  an  empirical  dissociation 
pressure  equation,  which  will  have  the  form  of  a  vapour- 
pressure  equation  (§  88). 
(iii.)  Method  of  mean  value  : 

dp  _  pz  —  pi 
dT  -  T2  -  T! 

where  TI,  T2  are  closely  adjacent. 
(Cf.  H.  M.,  §§  68—71.) 

Example.  —  Dissociation  of  ammonium  hydrosulphide  : 
NH.HS  -  -  >  NH3  +HS 

yas 


mm. 

T 

^cal 

.  atT 

175 

9-5  +  273 

24-500 

10-7  +  273 

212 

12-0 

21-730 

13-5 

259 

15-0 

24-090 

16-5 

322 

18-0 

20-490 

20-0 

410 

22-0 

22-500 

23-5 

501 

25-1 

dp_p»—pi  _  212-  175 
dT   T2  —  T!    12  -  9-5 

-  37 
2-5 

*»=£ 

CHANGES   OF  PHYSICAL   STATE  209 

p  =  $  (175  +  212)  =  193-5  mm. 
T  =  \  (282-5  +  285)  =  2S3'7° 


.-.  \a  =  2  X  1-985  X  X  14-8  =  24,500  cal. 

' 


A,,  is  nearly  constant,  the  mean  value  being  22,660  cal.     The 
value  calculated  from  thermochemical  data  is  22,800  cal. 
When  \d  is  constant,  (5)  may  be  integrated  : 

Inp  =  ^  +  const.        .         .         .         .  (6) 

which  shows  that,  so  far  as  pure  thermodynamics  will  take  us, 
Aj,  may  be  calculated  from  the  p,T  curve  : 


Pl 


(6a) 


~  T2 
although  the  converse  is  not  possible. 

In  general,  \e  depends  more  or  less  on  the  temperature,  and 
if  we  put  : 

\d  =  a  +  IT  +  cT2        .         .         .     (7) 
we  find  : 

Inp  =  A  +  BT  +  CM  +  ?     .         .         .     (8) 

where  A,  B,  C,  D  are  constants. 

Thus  there  is  found  for  the  system  : 

CaC03  ~  -  »  CaO  +  C02 
the  equation  : 


log  p  =  —    —  +  1-1  log  T  -  0-0012  T  +  8-882   mm. 


105.     Theorems  of  Robin  and  Moutier. 

Let  the  masses  1  —  m  and  m  of  two  phases  which  are  inter- 
convertible at  a  constant  temperature  and  pressure  with  absorp- 
tion or  emission  of  latent  heat  (e.g.,  water  and  steam,  or  ice  and 
water)  be  contained  in  a  cylinder  under  a  pressure  p  and  at  a 
temperature  T.  By  a  slight  motion  of  the  piston  let  a  further 
small  mass  8»j  of  the  second  phase  be  produced. 

If  $i(j>,T),  <£2(p,T)  are  the  specific  potentials  of  the  phases, 
the  increase  of  potential  of  the  system  is  : 


210  THERMODYNAMICS 

This  change  is  possible  and  irreversible  only  if 

(80  VT  <  0 
i.e.,  if  02  >  0i  then  5m  <  0,  i.e.,  the  second  phase  decreases  in 

amount  (e.g.,  condensation  occurs). 

if  02  <  0i,  then  8w  >  0,  i.e.,  the  second  phase  increases  in 
amount  (e.g.,  evaporation  occurs). 

> 
if  02  =  0i,  then  8m  =  0,  i.e.,  change  in  either  direction  is 

< 

possible  and  reversible,  and  the  system  is  in  equilibrium.  The 
condition  for  equilibrium  at  a  given  temperature  and  pressure  is 
therefore : 

02  (p,T)  -  0i(?>,T)  =  0         .         .         .     (1) 
i.e.,  the  specific  potentials  of  both  phases  are  equal. 

Since  0i(^),T),  02(j>,T)  depend  only  on  T  and  p,  and  since  these 
magnitudes  remain  constant  during  the  change,  it  is  evident  that 
the  total  potential  of  the  system  has  a  stationary  value  in  the 
equilibrium  state  ;  the  system  therefore  remains  in  equilibrium 
in  its  new  state,  and  the  state  of  equilibrium  is  neutral  (§  49). 

For  any  given  value  of  T,  equation  (1)  gives  on  solution  at 
least  one  value  of  p.  If  we  put  x  =  T,  y  =  p,  the  assemblage 
of  points  representing  the  various  possible  solutions  of  (1)  con- 
stitute a  curve  which  is  called  the  saturation  curve. 


dp"'  TOW  dp  -:t*foT)  •  •  •  (2) 
where  vi,  t-2  are  the  specific  volumes  of  the  two  phases  (V.r/.,  liquid 
and  vapour).  We  assume,  as  an  experimental  result,  that 

(3) 

<  0          j        .         .         .     (4) 

If  an  isotherm  T  =  TI  is  drawn  to  cut  the  saturation  curve, 
the  point  (or  points)  of  intersection  must  satisfy  (1),  i.e.,  in  this 
case : 

02(^)  —  <f>i(p)  =L  F(j>)  =  0  [T  const.]       .        .     (5) 

Since,  however,  dF(p)/dp  ^  0,   the    function   F(p)   has    no 

maximum  or  minimum  value,  hence  the  saturation  curve  cannot 
cut  the  isotherm  T  =  Tx  more  than  once,  and  since  this  holds 
for  all  values  of  T  which  satisfy  (1)  we  see  that  there  is  only  one 
value  of  p  corresponding  with  a  given  value  of  T. 


CHANGES  OF   PHYSICAL   STATE 


211 


If,  at  any  point,  ^  =  r2,  the  saturation  curve  can  exhibit  a 
maximum  or  minimum  (cf.  §  103). 

The  saturation  curve  divides  the  p,T  plane  into  two  regions, 
in  one  of  which  the  potential  of  the 
second  form  is  greater,  and  in  the 
other  less,  than  that  of  the  first 
form.  By  reason  of  the  sign  of 
$2  —  $1  ^6  call  these  the  positive 
and  negative  regions,  respectively. 

Let  AB  be  the  saturation  curve 
(Fig.  46), 

£>   -   01   =   0, 

and  let  P(T,p)  be  any  point  on  it. 

Through  P  draw  two  lines  aa, 

ftft'   parallel    to    the    axes    of   p    and    T    respectively. 
a  is  the  point  (T,  p  +  bp) 
ft  is  the  point  (T  +  ST,  p) 

and  a',  ft'  are  (T,  p  —  bp),  (T  —  8T,  p)  respectively. 
The  change  of  potential  in  passing  from  P  to  a  is  : 


FIG.  46. 


-  g?- 


where  i'i,  r2  are  the  specific  volumes  at  the  point  P. 

If  (v.2  --  vi)  >  0,  (^0)T  >  0,  and  a  lies  in  the  positive 
region.  The  only  change  which  can  spontaneously  occur  at  a 
is  one  which  makes  cm  <  0,  or  which  decreases  the  volume 
(e.g.,  condensation  of  steam)  when  it  occurs  on  the  saturation 
curve.  If  (r2  —  vi)  <  0,  then  (5<£}j.  <  0,  so  that  a  lies  in  the 
negative  region.  The  only  change  which  can  occur  spontaneously 
at  a  is  one  which  makes  bm  >  0,  i.e.,  which  decreases  the 
volume  (e.g.,  fusion  of  ice)  when  it  occurs  on  the  saturation 
curve. 

At  the  point  a'(bp  <  0)  the  directions  of  change  are  reversed, 
i.e.,  all  spontaneous  changes  increase  the  volume. 

These  results  are  summed  up  in  the  following  theorem,  due  to 
G.  Robin  : 

The  only  possible  spontaneous  transition  which  can  occur 
above  the  saturation  curve  (ftp  >  0)  is  one  which  leads  to 
diminution  of  volume  when  it  occurs  on  the  saturation  curve  ; 
the  only  possible  transition  which  can  occur  spontaneously  below 


212  THERMODYNAMICS 

the  saturation  curve  (8j>  <  0)  is  one  which  leads  to  increase  of 
volume  when  it  occurs  on  the  saturation  curve. 

In  all  cases,  the  change  within  the  equilibrium  system  which 
occurs  spontaneously  when  the  pressure  is  forcibly  altered  from 
without  is  one  which  tends  to  annul  the  alteration  of  pressure. 

The  change  of  potential  in  passing  from  P  to  /S  is  : 


where  sb  s2  are  the  specific  entropies,  and  L  the  latent  heat  of 
the  transition  [1]  —  »  [2]  ,  at  the  point  P. 

If  (*2  —  *j)  >  0,  Le.,  L  >  0,  then  §</>  <  0,  and  /3  lies  in  the 
negative  region.  The  only  change  which  can  occur  spontaneously 
at  /3  is  one  which  makes  fan  >  0,  or  which  absorbs  heat  when  it 
occurs  on  the  saturation  curve  (e.g.,  evaporation  of  water,  or 
fusion  of  ice). 

If  («2  —  «i)  <  0,  i.e.,  L  <  0,  then  5</>  >  0,  and  £  lies  in  the 
positive  region.  The  only  change  which  can  occur  spontaneously 
at  /3  is  one  which  makes  fan  <  0,  i.e.,  which  absorbs  heat  when 
it  occurs  on  the  curve. 

At  the  point  /3'(&T<0)  the  direction  of  the  spontaneous 
change  is  reversed,  i.e.,  it  occurs  with  evolution  of  heat  on  the 
curve. 

These  results  are  summed  up  in  the  following  theorem,  due  to 
J.  Moutier  : 

The  only  possible  spontaneous  transition  which  can  occur  in 
the  region  to  the  right  of  the  saturation  curve  (8T  >  0)  is  one 
which  leads  to  absorption  of  heat  when  it  occurs  on  the  satura- 
tion curve  ;  the  only  possible  transition  which  can  occur  spon- 
taneously in  the  region  on  the  left  of  the  saturation  curve 
(5T  <  0)  is  one  which  leads  to  evolution  of  heat  when  it  occurs 
on  the  saturation  curve. 

In  all  cases,  the  change  within  the  equilibrium  system  which 
occurs  spontaneously  when  the  temperature  is  forcibly  altered 
from  without  is  one  which  tends  to  annul  the  alteration  of 
temperature. 

Since  the  latent  heat  alters  only  slowly  with  temperature, 
Moutier's  theorem  can  be  applied  to  changes  not  too  far  removed 
from  the  curve  of  transition. 

Corollary  1.  —  Every  spontaneous  isopiestic  change  in  a  uni- 
variant  system  evolves  heat  if  it  takes  place  at  a  temperature 


CHANGES   OF   PHYSICAL   STATE  213 

lower  than  the  transition  temperature,  and  absorbs  heat  if  it 
occurs  at  a  temperature  higher  than  the  latter. 

Corollary  2. — If  there  are  two  opposite  isopiestie  transforma- 
tions possible  for  a  univariant  system  at  two  different  tempera - 
,tures,  the  one  occurring  at  a  lower  temperature  will  give  rise  to 
an  evolution,  that  at  the  higher  temperature  to  an  absorption 
of  heat. 

The  theorems  of  Moutier  and  Robin  apply  to  evaporation, 
fusion,  polymorphic  change,  or  dissociation  of  systems  in  com- 
pletely  heterogeneous  equilibrium. 

106.     Simultaneous  Equilibria  of  Physical  States. 

"We  have  still  to  consider  whether  it  is  possible  to  have  the 
three  forms  of  a  substance  coexisting  in  equilibrium  : 

[S]  —  [L]  z±  [G]   .         .        .        .     (1.) 

Again,  if  the  solid  can  exist  in  two  or  more  forms,  as  sulphur 
in  the  octahedral  and  prismatic  crystalline  forms,  there  are  the 
further  possibilities. 

[S-]  —  [Sp]  ^  [G]         .        .        •     C2> 
[SJ  ~  [S*]  -  [L]         .        .        .     (8) 

There  is  an  important  law  referring  to  such  equilibria,  which 
states  that  if  the  two  phases  A  and  B  of  a  substance,  and  the  two 
phases  A  and  C  are  at  a  given  temperature  in  equilibrium 
separately,  then  all  three  phases  will  be  in  equilibrium  together 
at  that  temperature.  Thus  if  two  phases  are,  at  a  given  tem- 
perature, separately  in  equilibrium  with  a  third  phase,  they  will 
be  in  equilibrium  with  each  other. 

We  shall  call  this  the  Law  of  Compatibility  of  Equilibria. 

This  result  is  an  immediate  consequence  of  the  potential 
equations.  Let  ^i(j*,T),  <j>^(p,T),  and  $3(j>,T)  be  the  potentials 
per  unit  mass  of  A,  B,  and  C,  respectively. 

If  A  and  B  are  in  equilibrium  at  a  pressure  o>  and  tem- 
perature -5  : 

^,(»,3)  =  0a(a>,d)     ....     (a) 

If  A  and  C  are  in  equilibrium  under  the  same  conditions : 

*i(«,*)  =  *«(«,*)    ....    (ft) 

.*.  from  (a)  and  (6) 


214  THERMODYNAMICS 

107.     The   Triple   Point. 

The  preceding  investigation  shows  that  the  T,p  curves  of  a 
substance  existing  in  three  phases  exhibiting  the  above  property, 
meet  at  a  point  which  has  the  specified  temperature  and  pressure 
as  co-ordinates.     This  is  called  the  triple  point  for  the  three  forms. 
Example.  —  Ice    and    water-  vapour    are    in    equilibrium    at 
+    0-0077°    C.,    under    a   pressure    of 
4'57    mm.,    and     liquid    water    is     in 
equilibrium   with   water-vapour   at  the 
same    temperature    and   pressure  ;  ice, 
liquid    water,    and    water-vapour     are 
therefore   in    equilibrium   under    these 
conditions,  and  the  equilibrium  curves 
representing  pressures  as  functions   of 
temperature    meet    at    a    triple    point 
?  (0-0077°  C.,  4-57  mm.).     These  curves 


FIG.  47. 

(i.)  OA,  the  evaporation  or  "vapour-pressure"  curve,  along 
which  liquid  and  vapour  are  in  equilibrium  ; 

(ii.)  OB,  the  sublimation-curve,  along  which  solid  and  vapour 
are  in  equilibrium  ; 

(iii.)  OC,  the  fusion-curve,  along  which  liquid  and  solid  are  in 
equilibrium. 

0  is  the  triple  point  ;  in  the  regions  AOB,  BOG,  COA,  homo- 
geneous vapour,  solid,  and  liquid  respectively,  are  stable  forms. 
The  curves  AO,  CO  may  be  prolonged  past  0  for  a  short  distance  ; 
these  prolongations  represent  supercooled  liquid,  and  superheated 
liquid,  respectively  ;  the  prolongation  of  BO,  which  would  repre- 
sent superheated  solid,  has  never  been  realised  (cf.  §  96).  We 
shall  prove  immediately  that  all  substances  of  the  ice-type  have 
curves  similar  to  the  above  ;  if  the  substance  is  of  the  wax-type, 
OC  slopes  to  the  right. 

The  existence  of  the  triple  point  was  first  indicated  by  James 
Thomson  (1851). 

108      Kirchhoff's   Formula  and   the   Equations  of   the  Triple 
Point. 

Regnault  (1847)  instituted  a  series  of  experiments  to  decide 
whether  the  vapour-pressure  of  the  solid  form  of  a  substance  was 
the  same  as,  or  different  from,  that  of  the  supercooled  liquid  at 


CHANGES   OF   PHYSICAL   STATE  215 

the  same  temperature.  He  concluded  that  "  the  passage  of  a 
body  from  the  solid  to  the  liquid  state  produces  no  appreciable 
change  in  the  curve  of  elastic,  force  of  its  vapour  ;  this  curve 
preserves  a  perfect  regularity  before  and  after  the  transition." 
This  implies  that,  if  the  crystalline  substance  and  the  supercooled 
liquid  were  contained  in  the  two  branches  of  an  inverted  U-tube, 
the  whole  would  be  in  equilibrium  at  a  given  temperature. 
Gernez  (1888),  who  made  the  experiment  with  acetic  acid,  found, 
however,  that  the  liquid  slowly  diminished  in  quantity ;  the 
amount  of  solid  at  the  same  time  increased. 

The  incorrectness  of  Regnault's  conclusion  was  demonstrated 
by  Kirchhoff  in  1858 ;  he  proved  that  the  vapour-pressure  curves 
of  solid  and  liquid  are  not  continuous  through  the  freezing-point, 
but  are  inclined  at  an  angle. 

Let  [1],  [2],  [3]  be  any  three  modifications  of  a  substance 
which  can  exist  together  in  equilibrium  at  a  triple  point,  and  let 
i'i,  i'2,  r3  be  their  specific  volumes  ;  «i,  «s-2,  *3,  their  entropies  pel- 
unit  mass.  The  gradients  of  the  p-T  curves  at  the  triple  point 
are  given  by  the  latent-heat  equations : 

(i)  d$  =  w^- 

al        l(t'3  —  ra) 

(2)  ^ L,_ 

dT       T(r3  -  n) 

(3)  (fy?3  __        L3 
f/T       T(ra  —  n) 

Where  p-t,  L,  denote  the  pressure  and  latent  heat  of  transition 
in  the  system  which  does  not  contain  the  i-th  phase. 
Since  the  changes  proceed  at  constant  pressure  (§  25) : 

L2  =  L!  +  L3       ....     (4) 

e.g.,  the  latent  heat  of  sublimation  =  latent  heat  of  fusion  + 
latent  heat  of  evaporation. 

Also  TI  —  v3  =  (r2  —  r3)  -4-  (n  —  r2)  identically  .  .  (5) 
and  LI  =  T  («8  —  sa)| 

L2  =  T(*8  -  *0[    ....     (6) 
L3  =  T  («2  -  Si)} 

By  elimination  of  L  and  v  from  (1)— (6)  we  obtain  the  funda- 
mental equations  of  the  triple  point : 


216 


THERMODYNAMICS 


(*  -  %)  ~  +  (*s  -  «i)  ~  +  («i-  *)  f£  =  0 
dpi  dp2  dp3 

which  may  also  be  written  in  determinant  form : 


(8) 


dT 


dT 


=  0 ;  and 


-7-  Si 

dpi 


dT 


If  1,  2,  3  refer  to  solid,  liquid,  and  vapour,  and  if  the  volumes 
of  solid  and  liquid  are  neglected  in  comparison  with  that  of  the 
vapour : 

dpe ~Le  t  dps L, 

dp,       dpr\  _  L,  (q} 

"     5fJ-W, 

an   equation   due   to   Kirchhoff  (1858),  which   shows   that   the 
difference  between  the  slopes  of  the  sublimation 
and  evaporation  curves  has  a  finite  and  positive 
C.P  value  at  the  triple  point,  so  that  the  sublima- 
tion  curve   lies   beneath   the   prolongation  of 
the  evaporation  curve.      The  gradient  of   the 
fusion  curve  near  the  triple  point  is  determined 
by  the  sign  of  (vt  —  r,),  and  according  as  this 
is  positive  (wax-type),  or   negative   (ice-type), 
the  curve   slopes  from   left  to   right,  or  from 
o  to  to  so  40  so  £  righi  to  left>  upwards.     The  difference  on  the 
FIG.  48.  left  of  (9)  is  usually  very  small. 

Thus  in  the  case  of  water,  Fischer  (1886)  found  0-0465. 
But  L/=  80  X  0-0413  1.  atm.  =  3'304  1.  atm. 

T  =273 

vg  =  209-905  litres  =  sp.  vol.  water- vapour  at  0°  C. 

•••(!•-  £) =2-f^=«™*<- 

=  0-0446  mm. 

This  very  small  value  had  eluded  Eegnault's  examination  of  his  curves, 
but,  as  Kirchhoff  showed,  it  can  be  inferred  from  his  experimental  data. 

The  sublimation  and  evaporation  curves  of  solid  and  liquid  phosphonium 
chloride  (PH4C1),  however,  meet  at  a  very  decided  angle  at  the  triple  point 
(Tammann,  Kryst.  und  Schmelz.,  p.  291).  The  curves  are  represented  in 
Fig.  48. 


CHANGES  OF  PHYSICAL   STATE  217 

Kirchhoffs  investigation  does  not  show  that  the  sublimation 
and  evaporation  curves  meet  each  other  at  the  temperature  at 
•which  solid  and  liquid  are  in  equilibrium  with  vapour  ;  it  proves 
that  they  are  inclined  at  an  angle,  but  the  further  fact  that  they 
intersect  requires  separate  proof,  which  was  inferred  by  James 
Thomson,  and  experimentally  demonstrated  by  Ferche  (1891)  in 
the  case  of  benzene  ;  the  point  of  intersection,  calculated  from 
the  vapour-pressure  curves,  was  5'405°  C.,  whereas  the  melting- 
point  was  5'42°  C. 

We  return  to  the  general  case. 

Let  ri  >  i  a  >  r3    .......  (9) 

and  write  (7)  in  the  form  : 


Then  from  (5),  (9),  and  (10)  we  see  that  -^  is  intermediate  in 

value  between  -^jj  and  (-jj.     This  implies  : 

Theorem  I.  (Moutier,  1876).—  If  an  isotherm  T  +  </T  is  drawn 
to  cut  the  three  curves  of  transition  (or  their  prolongations)  meeting 
at  a  triple  point,  the  central  point  of  section  corresponds  with  the 
transition  which  involves  the  greatest  change  of  volume,  or,  the 
latter  curve  of  transition  lies  between  the  other  two. 

Again,  let  «i  >  s2  >  s3  .......     (11) 

and  write  (8)  in  the  form  : 


*1  —  S3 

Then  from  (4),  (6),  (11),  and  (12)  we  find  Theorem  II.  :  Ij 
an  isopiestic  p  +  dp  is  drawn  to  cut  the  three  curves  of  transition 
(or  their  prolongations)  meeting  at  a  triple  point,  the  central  point 
of  section  corresponds  with  the  transition  involving  the  greatest 
change  of  entropy.  This  theorem  is  due  to  Eoozeboom  (1901). 

The  triple  point  divides  each  of  the  curves  of  transition  passing 
through  it  into  two  parts,  one  of  which  corresponds  with  a  stable 
system,  and  the  other  with  an  unstable  system.  The  discrimina- 
tion between  these  is  effected  by  means  of  two  theorems  due  to 
Roozeboom  (1887),  which  are  analogous  to  the  theorems  of 
Moutier  and  of  Robin,  for  two-phase  systems  (§  105). 


218  THERMODYNAMICS 

If  there  is  a  reversible  change  which  increases  the  entropy  of 
the  system,  but  leaves  its  volume  unchanged,  and  another 
reversible  change  which  increases  its  volume  without  alteration 
of  the  entropy,  there  will  in  both  cases  be  an  increase  in  the 
amounts  of  some  phases  and  a  diminution  in  the  amounts  of 
others. 

The  theorems,  III.  and  IV.,  in  question  assert  that,  if  the  mass 

of  thet-th  phase  -f  increases>  the  system  from  which  it  is  absent 
I  decreases 

cannot  exist  in  stable  equilibrium  at  : 

(1)  temperatures  {  J11^61', 
I  lower 


or  (2)  pressures 

(  lower 

respectively,  than  those  at  the  triple  point. 

Proof.  —  Let  the  system  have  the  temperature,  pressure,  volume, 
entropy,  and  potential,  T,  p,  Vi,  Si,  $1,  respectively. 

Keeping  T,  v,  and  </>  constant,  let  the  i-th  phase  be  caused  to 

appear  by  •]  \r         31.  "  the  entropy.     Then  for  the  second  state  we 
J  (  decreasing 

have 

$1   =   </>2          $1   —   $2 

Si  <  S2  or  Si  >  S2         .        .        .     (13) 
Vi  =  V2      Vi  =  V2 

Now  if  the  temperature  and  pressure  in  the  \  ,  state  are 

(  second 

changed  without   altering   the  relative  masses,  the  changes  of 

potential  are 

Sfa  =  -  SiST  +  ViSP        .  r     .        .     (14) 
Sfa  =  -  S2ST  +  V2SP        ,        .        %     (15) 

respectively,  and  if  i  ,  equations  (13),  (14),  and  (15)  lead 

(  ol  <  0 

at  once  to  : 

4>i  +  tyi  >  02  +  S</>2       .         .         .     (16) 
in   both   cases.      Thence,   since   in   stable   equilibrium   <£   is   a 

minimum,  it   follows  that,  at  a  temperature  slightly  j     ^ 

than,  and  under  a  pressure  differing  only  slightly  from  that  at 
the  triple  point,  the  system  containing  two  phases  cannot  be 


CHANGES  OF  PHYSICAL   STATE  219 

in    stable    equilibrium.     In    a    similar   way,    if    we    take    the 

relations : 

</>!  =   $2  </>!   =   $2 

Si  =  S2  and  Si  =  S2 

Vi  >  Va         Vi  <  Y2 

with  &p  >    0          Bp  <  0, 

we  obtain  again  the  inequality  (16),  and  thence  deduce  the  second 
theorem. 

This  investigation  is  due  to  P.  Saurel  (1902). 

We  now  return  to  the  equations  of  the  triple  point : 

(r,  -  !»  ^  +  (r.  -  n)  '&  +  (n  -  v2)^j  =  0, 
(*2  -  «.)  ^  +  (*8  -  «i)  ^  +  («i  -  *a)  ^  =  0, 

<//>!  f//>2  <fy>3 

and  suppose  that 

«'l   >   ''2  >    ''3 

«1  >  «2  >  S3, 

respectively. 

The  first  of  the  two  theorems  just  established  shows  that  the 
systems  of  two  phases  can  be  grouped  into  two  classes,  the 
stabilities  of  which  are  determined  by  the  signs  of  (r2  —  r3), 
(''a  —  t'i)i  (*'i  —  fa),  positive  coefficients  forming  one  class,  and 
negative  coefficients  the  other.  The  members  of  one  of  these 
classes  will  be  stable  at  temperatures  above,  those  of  the  other  at 
temperatures  below,  that  of  the  triple  point.  The  systems 
without  [1]  and  [3]  form  one  class,  and  that  without  [2]  the 
other.  Thus  we  have  the  Theorem  V.,  due  to  Duhern  (1891): 
The  system  of  two  2)nases  corresponding  with  the  transition 
involving  the  greatest  change  of  volume  is  in  stable  equilibrium  at 
temperatures  which  He  on  one  side  of  the  triple  point,  while  the 
other  two  systems  are  in  stable  equilibrium  at  temperatures  which 
lie  on  the  other  side  of  tlic  triple  point, 

The  second  of  the  two  theorems  shows  that  the  systems  of  two 
phases  can  be  grouped  into  two  classes,  the  stabilities  of  which 
are  determined  by  the  signs  of  (§2  —  #3),  (#3  —  si),  (*i  —  s2), 
positive  coefficients  forming  one  class,  and  negative  coefficients 
the  other.  The  members  of  one  of  these  classes  will  be  stable  at 
pressures  above,  those  of  the  other  at  pressures  below,  that  of  the 
triple  point.  The  systems  without  [1]  and  [3]  form  one  class, 
that  without  [2]  the  other.  Thence  we  deduce  the  Theorem  VI. 


220  THERMODYNAMICS 

(  Roozeboom,  1901)  that  the  system  of  two  phases  which  corresponds 
with  the  transformation  involving  the  greatest  change  of  entropy  is 
in  stable  equilibrium  under  pressures  lying  on  one  side  of  the  triple 
point,  while  the  other  two  systems  are  in  stable  equilibrium  under 
jwessures  lying  on  the  other  side  of  the  triple  point. 

Theorems  I.  and  V.,  or  II.  and  VI.,  lead  to  the  Theorem  VII., 
due  to  Gibbs  (1876)  :  If  a  small  circuit  is  drawn  around  the 
triple  point,  it  cuts  alternately  stable  and  unstable  branches  of  the 
curves  of  transition  meeting  at  that  point. 

Roozeboom  ("  Heterogcne  Gleichgewichte,"  I.,  189  (1901) ), 
has  shown  that  the  Theorems  I.,  II.,  V.,  VI.,  and  VII.,  together 
with  the  latent  heat  equations 

dpi  _  s2  —  s3  dfo  _  s3  — si    (fy?3  _  .§i  —  s2 
dT      i-a  —  v8'  f?T  ~~  r3  —  TI   dT       Vl  —  ra' 

furnish  a  complete  classification  of  the  various  possible  types  of 
triple  point. 

(B.  Roozeboom  :  Heterogen.  Gleidigewichte,  I.,  1901 ;  P.  Saurel, 
Journ.  Phus.  Chcm.,  1902;  P.  Duhein,  Zeitschr.  physik.  Cltem., 
8,  367,  1891  ;  Gibbs,  Scientif.  Papers,  I.) 


CHAPTER  Till 

VAN     DEB    WAALS'     EQUATION    AND     THE     THEORY    OF    CONTINUITY    OF 
STATES 

109.     Van   der   Waals'  Equation. 

In  Chapter  VI.  it  has  been  shown  that  the  characteristic 
equation  of  ideal  gases  : 

pr  =  ET (1) 

although  closely  followed  at  low  pressures,  is  deviated  from  more 
or  less  bj*  all  gases.  At  a  certain  point,  the  gas  may  pass  into  a 
liquid,  and  the  deviation  is  then  very  drastic.  The  critical  point 
is  also  entirely  left  out  of  consideration.  The  question  now 
arises  as  to  whether  it  is  possible  to  arrive  at  an  equation  which 
adequately  represents  the  behaviour  of  actual  gases,  including 
their  liquefaction,  and  critical  phenomena. 

Such  a  characteristic  equation  may  be  found  empirically,  or  by 
the  aid  of  theoretical  considerations  lying  outside  pure  thermo- 
dynamics. 

The  problem  has  been  largely  worked  at  from  both  sides; 
from  the  theoretical  side  the  point  of  view  has  been  almost 
exclusively  that  of  the  kinetic  gas  theory.  It  must  be  kept  in 
mind,  however,  that  it  is  possible  that  a  purely  mechanical  theory 
may  not  be  sufficient  to  cover  the  phenomena,  as  has  recently 
appeared  in  the  case  of  the  specific  heats  of  solids. 

The  first  characteristic  equation  to  be  proposed  which  gave  an 
adequate  representation  of  the  properties  of  gases  was  the 
equation  of  van  der  Waals,  which  resulted  from  a  revision  of  the 
deduction  of  the  equation  (1)  from  the  kinetic  theory,  and  the 
introduction  of  corrections  in  the  fundamental  assumptions 
that: 

(i.)  the  volume  of  the  molecules  themselves  is  negligible  com- 
pared with  the  total  volume ; 

(ii.)  there  are  no  forces  acting  between  the  separate  molecules. 

The  first  correction  leads  to  a  free  volume  (r  —  b)  instead  of  v ; 


222  THERMODYNAMICS 

the  second  to  an  active  pressure  uj  -f-  -^J  instead  of  p — the  latter 
being  diminished  by  the  attractive  forces ;  thence  : 

—       RT      _  <L  (2) 

M(r  —  b)       r2 

where  p,  v,  T  are  the  observed  pressure,  specific  volume,  and 
absolute  temperature,  R  is  the  gas  constant  (§  69),  and  a,  b  are 
constants  depending  on  the  chemical  composition.  According 
to  van  der  Waals,  b  is  four  times  the  actual  volume  of  the 
molecules  in  the  volume  v;  according  to  0.  E.  Meyer,  the 
multiple  is  4\/2  instead  of  4,  and  this  has  been  verified  by 
Heilborn  (1892)  and  by  Young. 

There  is  no  doubt,  however,  that  the  two  so-called  "constants," 
a  and  b,  are  functions  of  temperature,  and  perhaps  also  of 
pressure.  Numerous  attempts  have  been  made  to  express  this 
dependence,  particularly  that  on  temperature,  and  so  to  obtain 
more  exact  equations.  Clausius  (1880)  wrote: 
RT  /,  a'(v-b)\ 


., 

(3  constants> 

and  afterwards  : 

RT       /,       (a'T-n-  b')(v-b}\  f. 
2 


which  Battelli  (1892)  used  in  the  extended  form  : 

RT       fn       M(a'T  -  m  —  a"T  ~  n)  (v  —  6)~|  ,, 

J     (6  constants). 


The  need  for  so  many  constants  shows,  says  Weinstein 
(Thermodynamik  u.  Kinetik  dcr  Korper,  I.,  369),  "how  really 
unsatisfactory  is  yet  the  condition  of  this  branch  of  enquiry." 

Dieterici  (1899)  proposed  to  replace  r2  in  van  der  Waals' 
equation  by  r%  : 

,.       RT 


According  to  Boltzmann  and  Mache,  the  magnitude  b  is  not 
quite  independent  of  p  and  T  ;  at  high  pressures  it  becomes  only 
a  very  small  multiple  of  the  molecular  co-volume.  If  b  ^  is  the 
value  at  great  rarefaction,  they  proposed  the  equation  : 


VAN   DEE   WAALS'   EQUATION  223 

The  limits  of  variation  of  b  are  probably  not  greater  than  the 
ratio  1 :  10. 

(O.  E.  Meyer :  Kinetic  Theory  of  Oases  (trans.  Baynes).  Kuenen :  Die 
Zustandsghichung  der  Gasen  (1907).  Weinstein  :  Thermodynamik  and  Kinetik 
der  Korper,  vols.  1  and  2.  Boltzmann  :  Vorlesungen  tiler  Gastheorie.  J.  D. 
van  der  Waals,  Die  Kontinuitdt,  Leipzig,  1881.  Jeans  :  Dynamical  Theory 
of  Gases,  1904. 


110.     Thermal    Coefficients   from   van   der   Waals'   Equation. 

RT          a 

*  =  (7inr)-F    ...•(«) 

where  v  =  molecular  volume.  The  values  of  a  and  b  depend 
of  course  on  the  units  of  p  and  r.  Van  der  Waals  wrote  the 
equation  in  the  form  : 


by  analogy  with  the  equation  for  ideal  gases  : 

pv  =  jWo(l  +  aO)     (2)0  =  r0  =  1) 

and  hence  unit  pressure  =  1  atru.,  and  unit  volume  is  taken  as 
the  volume  of  1  kg.  at  N.T.P.  The  values  of  a  and  b  are  then 
called  the  "  constants  calculated  according  to  the  initial  volume 
as  unity." 

If  the  equation  is  written  in  the  form  (a),  a  and  b  refer  to 
1  mol  at  1  atm.  pressure. 

If  other  units  are  chosen  such  that  : 


a'  =  aa, 

//  =  pb, 

R'  =  /c 

Units 

a 

ft 

P 

cm.3,  atm. 

1 

I 

1 

cm.3,  mm.  Hg 

760 

1 

760 

litre,  atm. 

10~° 

io-3 

io-3 

litre,  mm.  Hg 

0-00076 

io-3 

0-760 

Thus  for  C02  Sundell  (1899)  gives  : 

1._  0-00525) 

=  (1  +  0-0129)  (1  —  0-00525)  (1  +  0'003660) 
=  1-0076  (1  +  0-003660) 
=  0-00369  (6  +  273) 
(Ideal  gas  :  pv  =  0'00366  (6  +  273).) 


224  THERMODYNAMICS 

In  the  case  of  methylamine  CH3NH2,  for  1  mol  as  unit : 
(a)  v  in  cm.3,  _?>  in  atm. : 

(P  +  74  *  1Q5)  (F  -  61)  =  82-09T 
(6)  r  in  litres,  p  in  mm.  Hg : 

L  -j_  5|^  (f  -  0-061)  =  62-39T. 
(Values  of  a  and  6  in  Winkelmann  :  Physik,  III.,  5,  857.) 

Ml)  •;=;CT<indePento'toIT)    •  <: 


,9T/  v  '  2?      Xv  ~~  &) 


/9p\               /      rT           2a\ 

•     \*> 
•     (3) 
•     (4) 

.  (4a) 
(Ah\ 

!k\  ^r')2   / 

_«p-     c»       T(w£jvm),       1         2a                2 

If  a  is  very  small  : 
1-6 

cp       cv-        -    273 

If  6  is  very  small  : 
r 

If  both  a  and  6  are  small : 


from   which   a   value   of  the   mechanical  equivalent  J  can  be 
calculated  (cf.  §  71). 

The  maximum  work  of  isothermal  expansion  is : 


.     (5) 
The  work  of  adiabatic  expansion,  if  we  put: 


p  +  -j'    (v  —  6)K  =  const. 


is  Att=^_(T1-TJ)-a^-^      .        .     (6) 


VAN  DEE  WAALS'   EQUATION  225 

The  Joule-Kelvin  effect  may  also  be  calculated  from  van  der 
Waals'  equation.  For  an  expansion  from  TI  to  r2  at  constant 
temperature,  let  the  change  of  intrinsic  energy  be  «2  —  HI  =  A»T. 

'   ' 


and 


a 


P  =  ^Tb~^ 


()ti\     a 

N-)r~  v2 


(9) 


•'•  "2  -  HI  =  A»T  =  I    ~  dv  =  a  (---- 

V  \Vi         ?2 

J    v, 

an  equation  also  deduced  on  purely  kinetic  grounds  by  Bakker 
(1888). 

In  the  Joule-Kelvin  experiment  : 

(MS  +  p*r*)  —  (MI  +  jpit'i)  —  A(w  +  pv)  =  Ait?  =  0. 
Suppose  the  change  from  the  state  before  the  plug  (p^,  rb  TI) 
to  that  after  (p%,  r2,  T2)  effected  in  two  stages  : 

(i.)  Expansion  at  constant  temperature  TI.     Then  if  n'  is  the 
intrinsic  energy  in  the  intermediate  stage  : 

/I       1 

u  —  HI  =  a  I  —  -- 
\vi      ra. 

(ii.)  Cooling  from  TI  to  T2  at  constant  volume  ra 

u2  =  u'  +  c,(T2  —  TO. 

.*.  cv(Ti  —  T2)  =  a  ^—  —  - 


r2  —  >         ii  —  w 
which  gives  the  cooling  effect  (rl\  —  T2). 

For  given  values  of  r2  and  TI  the  cooling  is  a  maximum  when 

[cc(Ti  -  T2)]  =  0 


which  gives  the  required  initial  volume.  Since  the  equation  is 
symmetrical  with  respect  to  n,  r2,  the  value  of  v2  for  maximum 
cooling,  with  given  i^  and  Tgjs  given  by  : 


226  THERMODYNAMICS 

The  cooling  effect  vanishes  when  Ti=T2,  .'. 
_2a  _  _        rTi&_ 
vira  ~  (ii  —  6)  (v2  —  b)' 

This  depends,  for  given  values  of  a,  1>,  Tb  on  both  Vi  and  r2. 
Since  r2  is  large  compared  with  b,  we  can  put  v%  /  (r2  —  />)  =  !, 
and  obtain  : 


If  the  initial  volume  is  less  or  greater  than  the  root  of  this 
equation  there  will  be  warming  or  cooling,  respectively.  We 
have  already  mentioned  that  the  inversion  temperature  of 
hydrogen  is  —  80°  C. 

The  equation  also  serves  to  calculate  the  deviation  of  a  gas 
from  Boyle's  law  : 

rTv         a 

pv  =  ----  7-  —  - 

v  —  b        r 

\  rT6       ,    a. 

du  1  T  ~       (v  —  b)*  "*"  v2 
and  the  sign  of  the  deviation  is  positive,  negative,  or  zero  according 

as  rT  is  less  than,  greater  than,  or  equal  to  j-  1  1  —    J  ,  in  which 

cases  the  gas  is  less,  or  more,  compressible  than  an  ideal  gas, 
or  in  the  latter  case  the  '  gas  behaves  on  compression  like  an 
ideal  gas,  although,  since  a  is  not  zero,  it  exhibits  a  Joule-Kelvin 
effect.  This  behaviour  is  exhibited  at  least  once  by  all  gases 
(except  perhaps  hydrogen),  viz.,  at  the  point  where  the  order  of 

compressibility  changes  sign.    The  maximum  value  ofy  (l  — 

is  ,,  when  r  =  oo.     If  >-T  is  greater  than,  less  than,  or  equal  to  ,, 

the  gas  is  for  all  densities  less  compressible  than  an  ideal  gas  ; 
or  is  more  compressible  on  one  side,  and  less  compressible  on  the 
other  side,  of  the  transition  point,  than  an  ideal  gas  ;  or,  at  the 
transition  point,  is  equally  compressible,  respectively.  These 
conclusions  agree  with  the  results  of  Amagat  (§  80). 

111.     Calculation  of  Critical  Constants   from   van   der   Waals' 
Equation. 

If  we  write  van  der  Waals'  equation  in  the  form  : 


p  p 


VAN   DEE  WAALS'   EQUATION 


227 


we  see  that  it  is  a  cubic  equation  in  v,  and,  for  every  given  pair 
of  values  of  p  and  T,  there  will  be  three  values  of  v,  because  a 
cubic  equation  has  three  roots.    Further,  the  theory  of  equations 
shows  that  these  roots  are  either : 
(i.)  all  real, 

or  (ii.)  one  is  real,  and  two  imaginary. 

The  physical  interpretation  of  this  result  is  that,  according  to 
the  conditions  of  pressure  and  temperature,  the  fluid  to  which 
the  equation  is  applied  can  exist  either  in  three  states  with 
different  specific  volumes  at  the  same  temperature  and  pressure, 
or  else  in  only  one  state  (imagin- 
ary roots  having  no  physical 
significance).  Case  (ii.)  corre- 
sponds to  a  (jas  heated  above  its 
critical  temperature.  In  case  (i.) 
the  physical  interpretation  is  that 
the  smallest  value  of  v  corresponds 
to  the  liquid,  the  largest  value  of 
v  corresponds  to  saturated  vapour, 
and  the  intermediate  value  corre- 
sponds to  an  unstable  state,  all  at 
the  given  temperature. 

These  relations  are  most  clearly 
seen  by  plotting  p  as  a  function  of  r  for  different  values  of  T 
(cf.  Fig.  49). 

The  curves  for  lower  temperatures  resemble  the  theoretical 
isotherms  of  J.  Thomson  (§  90).  The  real  isotherms  are  straight 
lines  abc,  Fig.  49,  along  which  two  phases  (liquid  and  vapour)  are 
present.  The  three  values  of  r  are  the  abscissae  of  a,  I,  c.  The 
values  at  a  and  c  correspond  with  liquid  and  vapour,  respectively, 
that  at  b  with  an  essentially  unstable  state,  for  there  the  isotherm 
slopes  from  left  to  right  upwards,  showing  that  the  pressure 
would  increase  along  with  the  volume,  i.e.,  the  elasticity  is  negative. 
It  must  be  observed  that  the  intermediate  value  does  not  refer  to 
a  heterogeneous  complex,  which  is  present  on  the  straight  line, 
for  van  der  Waals'  equation  holds  only  for  homogeneous  sub- 
stances. 

The  significance  of  the  portions  ad,  ec  of  the  theoretical 
isotherm  have  already  been  considered  in  connexion  with  rneta- 
stable  states. 

Q  2 


Fio.  49. 


228  THERMODYNAMICS 

The  points  on  the  various  curves  corresponding  to  a  and  c  are 
observed  to  approach  more  and  more  closely  as  the  temperature 
rises,  and  finally  they  coalesce.  At  this  point  the  three  roots 
become  identical 

rn  —  rb  =  r(.  =  rK,  say, 

and  the  point  of  inflection  thereby  indicated  is  the  critical  point 
of  the  substance.  z-K  is  the  critical  volume,  and  if  pK,  TK  are  the 
values  of  p  and  T  at  this  point,  these  are  the  critical  pressure 
arid  temperature. 

To  find  the  values  of  i\,  p^,  and  TK,  we  imagine  the  equation 
resolved  into  three  linear  factors : 


0  =  r3  — 

V  p  /    '      p        p 

where  a,  /3, 7  are  roots,  i.e.,  values  of  r  which  satisfy  van  der  Waals' 
equation  for  given  values  of  p  and  T.  At  the  critical  point  these 
are  equal : 


,-K  -|-  3n-K2  _  rR3 

=  !._,,|'^ +  ,,)+,«:_«:'' 
PK  I          PK       PK 


a       ,.,         1       8a 
or«K  =  84;ft  =  ^!  r»  =  -.^. 

Thus,  from  an  investigation  of  the  compressibility  of  a  gas  we 
can  deduce  the  values  of  its  critical  constants.  We  observe  that, 
according  to  van  der  Waals'  theory,  liquid  and  gas  are  really  two 
distant  states  on  the  same  isotherm,  and  having  therefore  the 
same  characteristic  equation.  Another  theory  supposes  that 
each  state  has  its  own  characteristic  equation,  with  definite  con- 
stants, which  however  vary  with  the  temperature,  so  that  both 
equations  continuously  coalesce  at  the  critical  point.  The 
correlation  of  the  liquid  and  gaseous  states  effected  by  van  der 
Waals'  theory  is,  however,  rightly  regarded  as  one  of  the  greatest 
achievements  of  molecular  theory. 

112.     Theorem  of  Corresponding  States. 

The  equation  of  van  der  Waals  leads  to  an  extensive  gene- 
ralisation as  to  the  relations  between  the  physical  properties  of 
various  substances. 


VAN   DER  WAALS'   EQUATION  229 

For  if,  in  the  equation 


the  pressure,  volume,  and  temperature  are  expressed  as  fractions 
of  the  critical  values 

p  =  *PI,  r  =  «K>  T  =  STK 
then 


But  rTK  =        ,  rK  =  36,  ;>K  = 


*  -l)  =  8d       .         .         .     (1) 

an  equation  from  which  all  constants  characteristic  of  the  specific 
nature  of  the  substance  (a,  b,  r)  have  disappeared.  Equa- 
tion (1)  is  in  fact  true  quite  generally  for  all  substances,  whatever 
be  their  chemical  nature,  provided  that  there  is  no  change  of 
molecular  complexity  (association  or  dissociation)  over  the  range 
of  pressures  and  temperatures  considered. 

TT,  <£,  d  are  called  the  reduced  pressure,  the  reduced  volume,  and 
the  reduced  temperature,  respectively,  and  equation  (1)  ma}'  be 
stated  in  the  form  that  if  we  know  the  critical  volume,  critical 
pressure,  and  critical  temperature  of  a  substance,  and  divide  the 
values  of  the  volume,  pressure,  and  temperature  in  a  series  of 
states  by  these,  the  quotients  will  satisfy  an  equation  which  does 
not  contain  any  constants  depending  on  the  specific  nature  of  the 
substance,  this  being  in  fact  the  equation  : 


Definition.  —  Any  states  of  two  substances  characterised  by  the 
same  values  of  TT,  <£,  3  are  called  Corresponding  States. 

Many  other  equations  of  state  besides  van  der  Waals'  lead  to 
laws  of  corresponding  states,  although  naturally  these  will  not  be 
of  the  form  (1). 

G.  Meslin  (1893)  has  investigated  the  conditions  under  which 
an  equation  of  state  can  lead  to  a  law  of  corresponding  states. 
The  most  general  form  of  equation  possible  is  : 

<J>(p,  r,  T,  d,  c2,  c9,  .  .  .)  =  0  .  .  .  (2) 
where  cif  c2,  c3,  .  .  are  constants  characteristic  of  the  particular 
substance. 


230  THERMODYNAMICS 

For  the  critical  point  we  must  have  (§  115)  : 


By  means  of  these  equations  we  can  eliminate  three  constants 
from  (2).  But,  if  the  equation  (2)  is  now  "  reduced,"  as  is 
required  by  the  law  of  corresponding  states,  it  must  not  contain 
any  constants  characteristic  of  the  substance,  hence  (2)  can  con- 
tain only  three  independent  characteristic  constants.  In  this 
case  (2)  can  always  be  written  in  the  form  : 

V<p,  r,  T,  pK,  rK,  TK)  =  0       .          .         •     (5) 

This  equation  must,  however,  be  independent  of  the  units 
adopted,  and  must  therefore  be  of  the  form  : 

0    ....     (6) 

and  since  this  contains  nothing  characteristic  of  the  nature  of 
the  substance  except  the  critical  constants,  it  is  a  reduced 
equation  of  condition. 

The  necessary  condition  that  an  equation  of  state  shall  lead  to 
a  law  of  corresponding  states  is,  therefore,  that  it  shall  contain  only 
three  characteristic  constants,  and  shall  exhibit  a  critical  point. 

As  an  example  of  the  application  of  the  reduced  equation  of 
condition  we  may  consider  the  expansion  when  any  substance  is 
heated  from  the  reduced  temperature  Si  to  the  reduced  tempera- 
ture $2  at  constant  reduced  pressure  TT  : 

</>2  -  </>i  =/(*,  32)-/(Oi) 

Divide  both  sides  by  <£i  and  put  <£  =  -r/rK  : 

~~  Vl 


i.e.,  the  fractional  expansion  is  the  same  for  all  substances  with 
the  same  range  of  •&  at  constant  TT. 


Corollary.  —  The  specific  volume  of  a  liquid  may  be  calculated 
from  (1)  at  all  temperatures  if  its  critical  temperature  and  its 
specific  volume  at  any  one  temperature  are  known,  by  comparison 
with  any  other  liquid  the  specific  volumes  of  which  are  known 
for  various  temperatures. 


VAN  DER   WAALS'  EQUATION  231 

As  standard  liquid  may  be  taken  fluorbenzene,  the  specific 
volumes  of  which  have  been  measured  by  Young  up  to  the  critical 
point  (d  =  560°  Abs.). 

113,    Evaporation. 

The  equation  of  van  der  Waals,  applying  as  it  does  to 
homogeneous  systems  only,  can  give  us  no  information  respecting 
vapour-pressures,  boiling-points,  etc.,  i.e.,  of  phenomena  charac- 
teristic of  heterogeneous  systems. 

In  such  cases,  however,  the  equation  can  be  combined  with  the 
theorem  of  Maxwell,  described  in  §  90.  If  we  consider  the  real 
and  fictitious  isotherms  of  evaporation,  ac  and  adbec,  we  have 
the  relation :  icork  along  real  isotherm  =  work  along  fictitious 
isotherm 

.-.  P(r9  -  r,)  =  I  pdv      ....     (1) 


•'9  —  r/)  =     1* 
J  '-i 


where  P  =  vapour-pressure  of  the  liquid,  and  p  =  /(*',T)  is 
defined  by  the  characteristic  equation.  If  we  adopt  van  der 
Waals'  equation  we  have 

/•T          a 

P  = 

...  p  (^  _  rf)  =  ,-T/w 
Corollary  1. — 

Vg    —     V 

i-T 

r   P~=T        ri 
since  both  a  and  c  lie  on  the  theoretical  isotherm. 
Corollary  2. — From  the  three  equations  we  can  find 

P  =  /(T),  Vg  —  ^(T),  r,  =  ^(T), 
and  thence,  by  the  Clausius-Clapeyron  equation 

L   =  T    /_  r)</P 

"  dT ' 

the  latent  heat  of  evaporation  can  be  obtained  (Dalton,  Phil.  Mag. 
13,  517,  1907). 
If  we  now  put 

a  =  3  ?>KrKa ;  b  =  l*,  from  §  111, 


232  THERMODYNAMICS 

in  (2),  and  divide  both  sides  by  ?r0,  we  get,  on  substituting  the 
reduced  magnitudes 

,  =  r/i-x,  *  =  T/TK, 

8  *,    302  —  1  ,„, 

o  3in  ZT-, —  .         .         .   .  lu  i 


in  which 

TT   =  reduced  pressure  of  saturated  vapour, 

0r  =  reduced  specific  volume  of  liquid, 

<£2  =         ,,          ,,  ,,       saturated  vapour. 

This,  with  the  equations  of  §  112,  gives  us  three  equations  of 
corresponding  states,  one  for  the  liquid,  one  for  the  saturated 
vapour,  and  one  for  the  heterogeneous  complex  : 


(1)  Liquid.  TT  +  (30X  _  1)  =  8$ 


(2)  Vapour.  TT  +      -2   (302  -  1)  =  8* 


(3)  Heterogeneous  complex. 

O         \  Q  Q/4»  1 

0102'  3  30i   —    1- 

from  which  we  obtain 

The  reduced  pressures,  and  specific  volumes  of  liquid  and 
saturated  vapour,  are  the  same  for  all  substances  at  equal 
reduced  temperatures. 

The  equation  TT  =  /(•&)  may  be  interpreted  thus  : 
If  for  two  different  liquids  the  reduced  temperatures  are  equal, 
so  also  are  the  reduced  vapour-pressures,  or  for  two  liquids  the 
ratios  of  the  vapour-pressure  to  the  critical  pressure  are  the  same 
if  the  ratios  of  the  temperature  to  the  critical  temperature  are 
the  same, 

PK\       PK.I      IKI        lK2 

As  an  illustration  we  will  take  Sajontschewski's  values  for  sulphur  dioxide 
and  ether : 

SOa  EtaO 

pK  =  78-9  atm. ;  TK  =  428-4  pK  =  36'9  atm. ;  TK  =  463. 

For  S02:  1'  =  60  atm.  when  T  =  412°'9 


=  0-964. 
428-4 


TAN   DER   WAALS*   EQUATION 


233 


For  Et2O :    P   =   «j>K  =   '7605    X   36'9  =   28-4  atm.  which  requires  a 
temperature  T  =  445 -8 

445-8 


463 


=  0-963. 


The  theorem  just  stated  may  be  written  in  the  form 


/     \ 
where  f  (  ~-  )  is  a  function  of  temperature  which  is  independent 

HK/ 

of  the  nature  of  the  substance.     Van  der  Waals  adopted  as  an 
approximate  equation 

.        .        .     (7) 


PropylAc 
PhF 


T 

or  log  P  =  —  a  ~Y  +  a  +  l°g  />K» 

in  which  a  is  a  constant  for  all  substances  and  is  approximately 
equal  to  3.  Guye  (1894)  has  calculated  the  values  of  a  for 
various  substances,  the  mean  is  :  a  =  3'06. 

In  the  case  of  alcohols,  acetic  acid,  and  water,  a  >  3'2,  indi- 
cating polymerisation  in  the  liquid  state. 

Bingham  (1906)  has  plotted  (  ™K  —  1 J  as  abscissa?  and  log^  as 
ordinates  for  the  substances  H2,  A,  Kr,  02,  CS2,  C6H5F,  Et20, 

CTT  rTinPTT         TTfOTT 
3ii6^v/L'^£l3,      HiUJXl. 

According  to  (6)  these  Log  S. 

should  give  coincident  £f0h 

straight  lines ;  Bing- 
ham found,  however, 
that  the  lines  not  only 
did  not  coincide,  but 
also  diverged  more  and 
more  as  the  absolute 
zero  was  approached. 
Nevertheless,  we  may 
still  conclude  that  the 
vapour-pressure  curves 
are  similar,  and  do  not 
intersect. 

The  curve  for  helium  lies  below  that  of  hydrogen.     It  is  at 
once  evident  that  although  the  deviations  from  the  theory  are 


FIG.  50. 


234  THERMODYNAMICS 

enormous  the}7  exhibit  striking  regularities.  The  higher  the 
molecular  weight  of  substances  belonging  to  the  same  series 
(e.g.t  He,  A,  Kr)  the  more  inclined  is  the  curve  to  the  axis  of 

(rp  \ 

-~  —  1  j  ,  and  the  same  result  also  follows  from  increasing  mole- 

cular complexity.  Thus  hydrogen,  although  it  has  a  molecular 
weight  less  than  that  of  helium,  is  more  complex  (H2),  and  its 
curve  is  steeper.  Valuable  conclusions  may  therefore  be  drawn 
from  such  curves. 

Corollary  3.  —  The   Clausius-Clapeyron  equation,   in  terms  of 
reduced  magnitudes,  may  be  written 

PK      d™  _  ^e          .  d"jr  _          L, 

TK  '  ~dd  ~  TK3rKFO)  '  '  d$~~  2>KrKdF(d)' 

OT>  rri 

But  psVK  =      Q  K,  and  hence 

8 

8  LeM. 

3WrK> 

and  since  the  left-hand  member  is  the  same  function  for  all 
substances,  it  follows  that 


i.e.,  the  quotient  of  the  molecular  heat  of  evaporation  at  any 
given  temperature  by  the  critical  temperature  is  a  constant,  or 
A(./T  is  the  same  at  equal  reduced  temperatures  for  all  substances. 

H20        Et20        (CH3)2CO        CHC13         CC14       CS2 
TT        7'5  1  1-41  1'49  1-57        2'03 

L,,       489  90  126-5  60  45  82 

1-35         1-31  1-44  1-35  1-34         T15 


Lemma.  —  The  boiling-points  under  atmospheric  pressure  are 
approximately  reduced  temperatures  (•&  =  §)  for  all  substances 
(Guldberg,  1890). 

Kurbatow,  for  carbon  compounds  with  more  than  5  carbon 
atoms,  finds  that  the  quotient  has  a  mean  value  of  0'666.  In 
homologous  series  it  varies  from  0'58  for  the  initial  members  to 
0'70  for  the  final  members. 

Corollary  4.  —  The  quotient  of  the  molecular  heat  of  evapora- 
tion by  the  absolute  boiling-point  under  atmospheric  pressure,  T0, 
is  a  constant  for  all  substances. 

This  rule  was  published  by  Trouton  ;  it  appears  to  have  been 


VAN  DER  WAALS'   EQUATION  235 

known  to  Despretz  and  to  Pictet  at  an  earlier  date.  The  va'ue  of 
the  constant  is  about  21;  according  to  Kurbatow  (1903),  the 
mean  value  is  20*7  ±  0*8.  If,  however,  the  constants  from 

van  der  Waals'  equation  are  substituted  in  the  equation  for  ™^, 

ID 

the  result  is  1O8,  which  is  widely  different  from  21. 
Xernst  (1906)  has  proposed  a  modified  rule  of  Trouton  : 

^=  9-5  log  TO  —  0-OOTTo. 

The  values  of  Ae/T0  vary  from  12*03  for  helium,  to  about  22 
for  substances  of  higher  molecular  weight  (aniline),  and  agree 
very  well  with  the  observed. 

According  to  Louguinin  (1900)  substances  associated  in  the 
liquid  state,  but  having  normal  vapour  densities  (alcohol,  water), 
exhibit  large  values  of  A,/T0,  whereas  those  associated  in  the 
vapour  (acetic  acid)  give  low  values.  If  the  molecular  weight  used 
in  estimating  \e  =  ML,,  is  taken  as  that  of  the  associated  vapour 
(e.g.,  (Ca'H^Oa^),  the  quotient  is  again  normal  if  the  liquid  is 
associated  to  the  same  extent.  This  is  the  case  with  acetic  acid. 

(Guye,  Arch.  Geneve  [3],  31,  163,  463,  1894 ;  Estreicher,  Phil  Mwj.  40, 
454,  1895;  Bingham,  Journ.  Amer.  Chem.  Soc.  28,  717,  1906;  Xernst: 
Recent  Applications  o/  Thermodynamics  to  Chemistry ;  Brill.  Ann.  Phys.  21, 
170,1906;  Guldberg.  Zntschr.  physik.  Chem.  5,  374,  1890.) 

By  analogy  with  Trouton's  rule  we  may  expect  that : 

(a)  The  molecular  heat  of  sublimation  of  a  substance,  divided 
by  the  absolute  temperature  at  which  the  sublimation  pressure 
is  equal  to  atmospheric  pressure,  is   approximately  constant : 
it  is  found  that  A,/T  =  30  (approx.). 

(b)  The  molecular  heat  of  dissociation  of  a  compound,  divided 
by  the  absolute  temperature  at  which  the  dissociation  pressure  is 
equal  to  atmospheric  pressure,  is  approximately  constant ;  it  is 
equal  to  32. 

These  rules  were  given  by  de  Forcrand  (1903).  Here  again 
the  agreement  is  only  approximate,  and  a  revised  rule  has  been 
proposed  by  Nerrist  (1906). 

^-  =  4-571(1-75  log  To  +  3-2). 

In  the  middle  of  a  fairly  extensive  range  of  temperature  ~- 
will  be  approximately  equal  to  32. 


236  THERMODYNAMICS 

Trouton,  Phil.  Mag.  [5],  18,  54,  1884;  de  Forcrand,  Ann.  Chim.  Phys. 
[7]  88,  384,  1903  ;  Kurbatow,  BeibL  28,  967,  1904  ;  Louguinine,  Ann.  Chim. 
Phys.  13,  289,  1898;  Arch,  de  Geneve  9,  5,  1900;  0.  £.  132,  88,  1901; 
Linebarger,  Sill.  Journ.  49,  380,  1895  ;  H.  Crompton,  Trans.  Chem.  Soc.  17, 
365,  1895,  D.  Berthelot,  Ann.  Chim.  Phys.  [7],  4,  133,  1895;  J.  Traube, 
Bar.  31,  1562,  1898;  G.  Bakker,  Ztitschr.  physik.  Chem.  18,  519,  1895; 
47,  231,  1904  ;  Batschinski,  ibid.  43,  369,  1903  ;  H.  von  Jiiptner,  ibid.  63, 
355,  579,  1908;  64,  709,  1908;  73,  173,  1910. 

Corollary  5.  —  The  equation  of  state  of  an  ideal  gas  is 

pv  =  RT, 

where  ?  is  the  molecular  volume,  R  =  8'26  X  107  inC.G.S.  units. 
If  we  assume  that  this  holds  up  to  the  critical  point, 

v  =  v',  T  =  TK,  and  V  =  ?^. 
PK 

But,  on  van  der  Waals'  theory 

01  a     r  1       8a 


_  3  RTK 
'  "K  "8  j* 

If  d,b  are  the  densities  of  the  vapour  at  the  critical  point 
calculated  on  the  laws  of  ideal  gases  and  van  der  Waals'  equation, 
respectively,  and  M  is  the  molecular  weight, 
d  =  MrK',  a  =  MrK 

^  Q 

.'.-7=5  =  2'67  for  all  substances. 

li  O 

Young  has  found,  however,  that  the  ratio  of  the  observed 
critical  density  to  that  calculated  on  the  laws  of  ideal  gases  is 
approximately  3*75  for  all  substances  except  those,  like  acetic 
acid,  which  are  polymerised. 

Dieterici  (1895)  has  observed  that  this  ratio  would  not  be 
less  than  3  if  6  were  assumed  to  be  a  function  of  the  volume  ;  on 
the  assumption  of  the  constancy  of  l>  he  has  given  two  new 
equations  of  state  which  give  values  of  bjd  closely  approximating 
to  the  observed  numbers 

(i.)  (  p  +  J  )  (r  -  fc)  =  RT  ;  8/rf  =  3-75 

(ii.)      p(v  —  b}  =  RT  .  .  .  e  ~  ^  ;  8/rf  =  3  "695. 
D.  Berthelot  (Mem.  Bureau  des  poids  et  mesures,  13,  Paris, 
1907),  has  proposed  a  modification  of  Clausius's  equation  which 


VAN  DER   WAALS'   EQUATION  237 

agrees  very  closely  with  the  compressibility  data  for  gases  up 
to  about  5  atm.  : 


-I-  J!U  (v  —  6)  =  RT, 
where  I  =  v~ 

o 

a  ~  64  R2  ~pl' 

Planck  (Be/7.  .Be;-.,  1908,   633)  has,  from   the   fundamental 
statistical  definition  of  entropy,  deduced  the  equation : 
RT  ,     {.        Bt\        a 


where  fi  =  26,  a  =  const,  which  differs  from  van   der  Waals' 
only  in  terms  of  the  second  order. 

114.     Testing  the  Theorem  of  Corresponding  States. 

Some  consequences  of  the  theorem  of  corresponding  states 
have  already  been  considered  with  reference  to  experimental 
results ;  it  has  been  shown  that  there  is  a  good  general  agree- 
ment, but  this  is  not  strict.  The  question  arises  as  to  whether 
the  deviations  observed  are  due  to  the  errors  of  experiment,  or 
are  indications  of  an  inherent  fault  in  the  equation  itself. 

Amagat  (1896)  made  an  ingenious  test  of  the  theorem  in  the 
following  manner.  If  the  isotherms  for  various  substances  are 
drawn  in  a  diagram  in  which  reduced  volumes  (V/L-K  =  </>)  are 
taken  as  abscissae  and  reduced  pressures  O>/J>K  =  TT)  as  ordinates, 
then  isotherms  having  the  same  reduced  temperature  must 
coincide,  and  the  whole  series  of  isotherms  must  appear  as  if 
they  represented  a  single  substance,  i.e.,  they  must  be  similar 
and  must  not  intersect.  Instead  of  using  reduced  values,  Amagat 
took  simply  the  ordinary  values  of  p  and  r  for  two  substances, 
drawing  one  set  of  curves  on  transparent  glass,  and  the  other  on 
paper,  and  then,  by  means  of  a  beam  of  parallel  light,  projected 
the  former  on  the  latter.  By  a  suitable  rotation  of  the  trans- 
parent diagram  about  either  axis,  the  relative  proportions  of 
abscissae  and  ordinates  could  be  reduced  directly,  and  it  was 
possible  to  determine  if  the  curves  could  be  made  to  form  a  non- 
intersecting  series.  This  was  the  case  for  air,  ether,  carbon- 
dioxide,  ethylene,  and  isopentane. 

Raveau  (1897)  adopted  an  even  simpler  method.    The  logarithms 


238  THERMODYNAMICS 

of  the  volumes  and  pressures  were  used  as  co-ordinates,  and 

log  —  =  log  v  —  log  rK  =  log  r  —  const., 

% 

and  similarly  for  p  ;  hence  the  curves  can  be  made  to  correspond 
by  merely  changing  the  position  of  the  origin,  i.e.,  by  motions 
of  one  diagram  over  the  other  parallel  to  the  axes.  Again  the 
theorem  was  found  to  hold  good. 

Raveau  now  calculated  the  values  of  p,  v  from  van  der  Waals' 
equation,  plotted  the  logarithms,  and  compared  the  diagram  with 
a  similar  one  drawn  from  the  experimental  results.  The  results 
showed  that  the  diagrams  could  not  be  made  to  fit  in  the  case  of 
carbon-dioxide  and  acetylene,  the  divergencies  being  very  marked 
near  the  cribical  point. 

The  results  of  Amagat's  and  Raveau's  work  may  be  summed 
up  in  the  statement  that,  whereas  the  theorem  of  corresponding 
states  holds  good  very  approximately,  the  equation  of  van  der 
Waals  gives  results  quite  inconsistent  with  the  experimental 
values,  especially  near  the  critical  point. 

There  still  remains  for  consideration  the  question  whether  the 
theorem  of  corresponding  states,  which  we  have  seen  is  at  least 
approximately  true,  is  in  fact  rigorously  exact,  or  is  only  a  more 
or  less  close  approximation.  This  problem  is,  thanks  to  the  now 
classical  investigations  of  S.  Young  and  his  students,  quite  satis- 
factorily solved.  Very  careful  measurements  have  shown  that 
there  are  small  deviations,  the  magnitude  of  which  is  much 
greater  than  the  experimental  errors,  and  the  theorem  of  corre- 
sponding states,  in  the  form  previously  employed : 

/(»,$,  3)  =?  6, 
where /is  the  same  function  for  all  substances,  and 

*  =  P!P*>  </>  =  '•/'•*  a  =  T/TK, 

is  not  rigorously,  but  only  very  approximately,  true. 

Madame  Kirstine  Meyer  (1900)  has  shown  that  the  discrepancies  are  not 
to  be  explained  by  errors  in  the  critical  data ;  the  law  of  corresponding 
states  can  be  tested  without  making  use  of  these  constants,  and  differences 
between  the  observed  and  calculated  magnitudes  are  still  apparent.  D. 
Berthelot  (Journ.  de  Phys.,  1903)  has  deduced  some  new  equations. 

The  results  of  Young,  and  others,  have  shown  that  substances 
may  be  divided  into  two  large  groups  according  as  they  do  or  do 
not  agree  closely  with  the  theorem  of  corresponding  states. 
These  may  be  called  "  normal  "  and  "  abnormal  "  substances, 


VAN   DEE   WAALS'  EQUATION  239 

respectively.  To  the  first  group  belong  the  hydrocarbons,  esters, 
ketones,  ethers,  etc. ;  to  the  latter  belong  water,  alcohols,  and 
fatt}'  acids.  The  monatomic  gases  such  as  helium  and  argon 
occupy  a  special  group,  the  members  of  which  agree  amongst 
themselves  but  not  with  other  normal  substances. 

The  normal  substances,  however,  really  exhibit  small  deviations 
which  are  all  the  greater  the  more  complex  is  the  molecule  of  the 
substance.  The  theory  of  van  der  Waals,  or  in  fact  any  hypothesis 
from  which  a  theorem  of  corresponding  states  could  be  derived, 
assumes  however  that  the  transition  from  the  gaseous  to  the 
liquid  state,  as  well  as  the  changes  of  density  in  either  state, 
result  from  alterations  in  the  propinquity  of  molecules  which 
otherwise  remain  unaltered.  Any  association  or  dissociation  of 
the  substance  would  therefore  give  rise  to  abnormalities,  and  in 
fact  the  substances  which  deviate  most  from  the  normal  relations 
(e.g.,  water,  acetic  acid)  are  those  which  appear,  on  other  grounds, 
to  be  associated  in  the  liquid  state.  In  the  case  of  acetic  acid 
the  commencement  of  polymerisation,  even  in  the  state  of  vapour, 
is  evident  from  the  abnormal  densities. 

A  similar  explanation  may  account  for  the  slight  deviations 
exhibited  by  "  normal "  substances,  but  fails  to  explain  the 
anomalous  behaviour  of  the  monatomic  gases.  A  mechanical 
interpretation  of  the  theorem  of  corresponding  states  has,  how- 
ever, been  advanced  by  Kamerlingh  Oniies  ("  Principle  of 
Uniformity  ")  which  appears  to  embrace  all  known  cases. 

In  short,  we  may  say  that  there  is  evidence  that  the  deviations 
from  the  relations  predicted  by  the  characteristic  equations  may 
be  due  to  chemical  changes  in  the  substances,  which  are  not 
taken  into  consideration  in  the  kinetic  deduction  of  the  equations. 
Weinstein  (loc.  cit.)  considers  that  it  is  possible  to  deduce  an  equa- 
tion which  takes  account  of  these  chemical  changes  as  well.  It  will 
be  sufficient  here  to  re-emphasise  the  fact  that  there  is  at  present 
no  characteristic  equation  known  which  agrees  accurately  with 
the  behaviour  of  a  single  substance,  let  alone  various  substances, 
over  a  wide  range  of  temperature. 

Amagat,  O.K.  123,  30,  83,  1896;  Journ.  de  Phys.  [3],  6,  1,  1897.  Eaveau, 
C.R.  123,  109,  1896 ;  Journ.  de  Phys.  [3],  6,  432,  1897.  K.  Meyer,  Zeitvchr. 
physik.  Chem.  32,  1,  1900.  D.  Berthelot,  C.E.  130,  565,  713,  1900;  131, 
175,  1900 ;  Journ.  de  Phys.  [3],  10,  611,  1901 ;  [4],  2, 186,  1906.  Kamerlingh 
Dimes,  Arch.  Ne'erl.  30,  128,  1896. 


240  THERMODYNAMICS 

115.     Gibbs's  Thermodynamic  Model. 

If  in  a  rectangular  left-handed  co-ordinate  system  with  the 
^-axis  upwards  we  put  : 

x  =  r,  y  =  S,  z  =  U 

for  the  whole  volume,  entropy,  and  energy  of  a  substance  in  its 
different  states,  the  aggregate  of  states  will  form  a  surface  which, 
since  it  was  first  described  by  Willard  Gibbs  (1873,  Scient. 
Papers,  L,  32),  is  called  Gibbs's  ThermoJijnamic  Model. 

If  we  draw  three  planes  perpendicular  to  the  axes  of  r,  S,  U, 
respectively,  these  will  be  the  loci  of  all  states  which  have  a 
constant  value  of  these  magnitudes,  respectively.  The  plane 
v  =  0  is  evidently  fixed,  but  the  planes  S  =  0,  U  =  0  can  only 
be  fixed  by  arbitrary  choice  of  the  initial  states  of  zero  entropy 
and  energy.  The  origin  may  therefore  be  chosen  anywhere  in 
the  plane  of  zero  volume. 

If  the  energy  is  taken  as  a  function  of  the  volume  and  entropy, 
we  have  : 


rfU=T<?S-j)rfr         ....     (2) 


An  equation  in  which  the  entropy  of  a  homogeneous  fluid  is 
expressed  as  a  function  of  its  energy  and  volume  is  called  by 
Planck  (1909)  a  canonical  equation. 

Equations  (3)  are  the  conditions  of  equilibrium. 

The  condition  for  stability  requires  that  : 

SU  >  TSS  -  p&r   .         .         .         .     (4) 

in  which  the  variations  are  to  be  construed  strictly,  i.e.,  quantities 
of  order  higher  than  the  first  are  not  to  be  neglected.  This  is 
equivalent  to  (2)  of  §  51. 

Now  if  we  pass  from  one  point  (r,  S,  U)  on  the  surface  to  an 
adjacent  point  (r  +  6r,  S  +  5S,  U  +  8U)  we  shall  have  : 


+     •         •     (5) 

Thus,  from  (3),  (4)  and  (5)  we  have,  as  the  condition  for  stable 
equilibrium  of  a  homogeneous  phase  : 


VAN  DER  WAALS'   EQUATION  241 

the  conditions  for  which  inequality  are  : 

I  ^F      ^P 

TO        ?U      !>°<6>:ggJ>°       '         •         •     <7> 


and  hence  ^  >  0 (8) 

Inequalities  (6)  and  (7)  show  that  the  surface  is,  at  every  point 
which  corresponds  with  a  homogeneous  phase  in  stable  equilibrium, 
convex  downwards  in  every  direction. 

Let  us  next  consider  the  equilibrium  of  a  heterogeneous 
complex  of  two  phases,  numbered  1  and  2,  of  a  single  component. 
The  conditions  for  equilibrium  are  : 


.     ..(11) 

Now  the  equation  of  the  tangent  plane  at  the  point  1  can,  as 
we  know  from  solid  geometry,  be  obtained  by  writing  the  sub- 
script 1  after  each  quantity  in  equation  (1)  and  then  putting  : 
dUi  =  U  -  Ux 

dSi  =  S  -  S! 

" 


From  (9)—  (12)  we  deduce  at  once  that  the  tangent  planes  at 
the  points  1  and  2  are  coincident.  Hence  the  theorem  : 

If  two  different  states  can  exist  permanently  in  contact,  the  points 
representing  these  states  on  the  thermodynamic  model  have  a 
common  tangent  plane. 

The  converse  is  also  readily  proved  : 

If  two  points  on  the  surface  have  a  common  tangent  plane,  the 
states  represented  by  them  are  such  as  can  exist  permanently  in 
contact  (cf.  §  53). 

We  can  therefore  tell  beforehand  whether  two  given  states  of  a 

T.  R 


242  THERMODYNAMICS 

fluid  will  be  in  stable  equilibrium  when  placed  in  contact, 
because  U  and  S  can  be  determined  by  some  process  in  which 
the  two  states  never  appear  simultaneously,  e.g.,  by  a  continuous 
transition  of  state  (§  87). 

Let  Mi,  Ui,  ri,  Si,  and  M2,  U2,  r2,  S2  be  the  masses,  specific 
energies,  volumes,  and  entropies,  of  the  two  phases,  respectively, 
then,  if  the  letters  without  suffixes  refer  to  the  whole  values  for 
the  complex  : 

M  =  M!  +  M2 
MU  =  MiUi  +  M2Ua 
MS  =  MA  +  M2Sa 
M.V  =  Mii?i  +  M2r2 

-j-  MaUa  \ 


MI  +  M2 

_  MiSi+M2S2 
—  -TTF  —  T^TF  — 
MI  -f-  M2 

Mara 


--  Tjjf  —  r~Tur  —       .... 

MI  +  M2 

so  that  the  representative  point  of  the  complex  lies  on  the  join 
of  the  points  1  and  2,  and  divides  this  line  into  two  segments 
which  are  inversely  as  the  masses.  Thence,  if  the  two  phases 
exist  in  equilibrium  at  a  given  temperature,  and  pressure,  there  is 
possible  a  continuous  series  of  states  of  equilibrium,  characterised 
by  constancy  of  the  specific  volumes,  entropies  and  energies  of 
the  separate  phases,  whilst  the  masses  of  these,  and  hence  the 
specific  volume,  entropy,  and  energy  of  the  whole  system,  change 
in  a  continuous  manner.  If  the  change  proceeds  from  one 
extremity  of  the  line  to  the  other,  we  shall  have,  from  (2)  : 

U2  -  Ui  =  T(S2  -  SO  -  X'-a  -  n)    -         -         -         -  (16) 

Now  suppose  the  representative  point  had  passed  from  1  to  2, 

not  along  the  line  of  heterogeneous  states,  but  along  some  path 

on  the  surface  itself;  this  will  be  a  continuous  transition  of  state. 

Then,  from  (1)  : 


If  the  path  is  an  isotherm  : 

au     /8u\      /au 


VAX   DER   WAALS'  EQUATION 


243 


.-.  U2  -  Ui  =  T( S2  -  SO  + 


and  if  the  path  is  an  isopiestic  : 

8U_  /am  _  /am  _ 

dr  ~  \dv  )  i  ~~  \af/2~ 

r2 

then  U2  —  Ui  =    I   ^  rfS  —  Xra  —  ri) 

|     £jj^ 

j  i 

Thence,  from  (16),  (19),  and  (20) : 


const.  =  — p  . 


9U 


Q  =  T(S2  -  Si)  = 


•  (19) 

•  (20) 
.  (21) 

•  (22) 
.  (23) 


which  may  be  regarded  as  the  analytical  expressions  of  the  two 
forms  of  Maxwell's  theorem,  concerning  the  theoretical  isotherm 
of  James  Thomson  (§  90). 

Since  the  energies  are  single-valued  functions  of  the  volumes 
and  entropies  of  the  substance  in  the  two  states,  and  since  the 
equations  (9),  (10),  (11)  give  three  relations  between  the  four 
quantities  rlf  Si,  t'2,  S2,  it  is  evident  that  if  one  is  fixed  the  others 
are  determined,  so  that  the  two  points  lie  on  two  definite  curves 
on  the  model : 

Ui  =,/iO'i,  SO,  and  U2  =/2(r2,  S2) 

and  to  every  point  on  the  one  there  is  a  single  corresponding 
point  on  the  other.  If  now  we  imagine  the 
doubly-tangent  plane  to  roll  along  the  surface, 
the  locus  of  its  points  of  contact  with  the  A 
surface  will  trace  out  two  curves,  called 
together  the  connodal  curve,  and  the  tangent 
plane  being  everywhere  tangent  to  the 
surface  representing  heterogeneous  states 
along  a  line,  it  follows  that  the  latter  (called 
by  Gibbs  the  derived  surface,  as  distinguished 
from  the  primitive  surface  representing  homo- 
geneous states)  will  be  a  developable  surface, 
and  will  form  a  part  of  the  envelope  of  the  successive  positions 
of  the  rolling  tangent  plane. 

Let  Pb  P2  (Fig.  51)  be  the  points  of  contact  of  the  tangent 

K  2 


FIG.  51. 


244  THERMODYNAMICS 

plane,  and  let  planes  through  these  points  perpendicular  to  the 
axes  of  v  and  S  respectively  intersect  in  the  line  AB,  which  will 
be  parallel  to  the  axis  of  U.  Let  the  tangent  plane  cut  this  line 
in  A,  and  let  PiB,  P2C  be  drawn  perpendicular  to  AB  and  parallel 
to  the  axes  of  v  and  S. 


. 

and  if  we  roll  the  tangent  plane  through  an  infinitesimal  angle 
about  the  instantaneous  axis  PiP2,  so  that  it  meets  AB  in  A', 
then  : 


dp  _  BP2  _  S2  —  Si  _         L 

"  dT  ~  CPl    -  i-2  —  1-1  ~  T(ra  -  n) 

which  is  the  Clapeyron-Clausius  equation. 

As  the  tangent  plane  rolls  on  the  primitive  surface,  it  may 
happen  that  the  two  branches  of  the  connodal  curve  traced  out 
by  its  motion  ultimately  coincide.  The  point  of  ultimate  coinci- 
dence is  called  a  plait  point,  and  the  corresponding  homogeneous 
state,  the  critical  state. 

The  conditions  which  must  be  satisfied  at  the  plait  point  may 
be  deduced  as  follows :  Expand  by  Taylor's  theorem  the 
expressions  on  the  right  of  (9)  and  (10),  omitting  terms  of  higher 
orders  than  the  second  : 

f<®\  _L  /95A  /q   _  o      ,   /  82U  \ 

\8s2/ 1(  2   '  l}  +  UsaJ  !(ra  ~ ri) 
aau\  „  ,  /a2uN 


and  hence,  when  r2  —  ri,  S2  —  Si  become  very  small  with  approach 
to  the  plait  point  : 

(S,-  SO 


and  at  the  plait  point  itself: 

a2u         82u 

T  .     S2  -  Si     . 


VAN  DEE  WAALS'  EQUATION  245 

or,  if  we  put  : 


cPU         3*0 

as2 


('25) 


the  equation  of  the  plait  point  is 

A  =  0  .....     (26) 

If  we  imagine  a  line  drawn  on  the  primitive  surface  dividing  all 
parts  of  the  surface  which  are  convex  downwards  in  all  directions 
from  those  which  are  concave  downwards  in  one  or  both  direc- 
tions of  principal  curvature,  this  curve  will  have  the  equation  (26), 
and  is  known  as  the  spi  nodal  curve.  It  divides  the  surface  into 
two  parts,  which  represent  respectively  states  of  stable  and 
unstable  equilibrium.  For  on  one  side  A  is  positive,  and  on 
the  other  it  is  negative.  If  we  assume  that  the  tie-line  of 
corresponding  points  on  the  connodal  curve  is  ultimately 
tangent  to  that  curve  at  the  plait  point,  it  follows  that 
the  direction  of  this  tangent  is  given  by  either  of  the  two 
equations  : 


Now  the  plait  point  is  on  the  spiuodal  curve,  and  any  two 
corresponding  points  of  the  coimodal  curve  adjacent  to  the  plait 
point  are  on  a  part  of  the  surface  which  is  convex  in  every 
direction,  and  for  which  therefore 

A>0. 

Thus  the  spinodal  curve  does  not  cut  the  connodal  curve  at  the 
plait  point,  and  it  is  simplest  to  assume  the  two  curves  to  be 
tangent  at  that  point.  From  (26)  it  follows  that  the 
direction  of  the  tangent  at  any  point  of  the  spinodal  curve  is 
given  by  : 


At  the  plait  point  this  equation  will  give  the  same  value  for 
as  equations  (27)  and  (28),  and  hence  at  that  point  the 


246  THERMODYNAMICS 

three  determinants  of  the  second  order  formed  from  the  matrix  : 
eW          a2U 

as2      asa<- 

a2u       a^u  , 

'      ' 


will  be  zero.  Again,  the  limiting  position  of  the  line  joining 
corresponding  points  of  the  connodal  curve  and  the  direction  of 
the  common  tangent  to  the  connodal  and  spinodal  curves  at  the 
plait  point  is  given  by  : 

a2!!         aau         aA 

T.      Sa-Si       rfS  aS8r  3r2  dv 

Lim  7^7  -  rff  -    -  w  =    ~W=    ~  aA  '    (31) 
as2         asar        as 

Conditions  (30)  and  (31)  are  sufficient  to  discuss  the  principal 
properties  of  the  critical  state  of  a  one-component  system.  We 
observe  that  the  existence  of  a  critical  state  for  such  a  system 
cannot  be  inferred  from  a  priori  considerations,  because  it  is  not 
necessary  that  the  two  branches  of  the  connodal  curve  should 
ultimately  coalesce  ;  that  such  is  the  case  must  be  regarded  as 
established  for  systems  containing  liquid  and  vapour  by  the 
experiments  of  Andrews  (§  86),  and  the  following  discussion  is 
limited  to  such  systems  (cf.  §  103). 

With  motion  along  the  connodal  curve  towards  the  plait  point 
the  magnitudes  Ui  and  Ua,  Si  and  82,  and  TI  and  r2,  approach 
limits  which  may  be  called  the  energy,  entropy,  and  volume  in 
the  critical  state.  The  temperature  and  pressure  similarly  tend 
to  limits  which  may  be  called  the  critical  temperature  and  the 
critical  pressure.  Hence,  in  evaporation,  the  change  of  volume, 
the  change  of  entropy,  the  external  work,  and  the  heat  of 
evaporation  per  unit  mass,  all  tend  to  zero  as  the  system 
approaches  the  critical  state  : 

r.2  —  TI  =  0 

S2  -  Si  =  0 

PK(VZ  -  »i)  =  0 

TK(S2-Si)=0 

Q         _     O 

The  ratio  --  --  -  will,  however,  in  general  tend  to  a  finite 

'a  —  i-i 
limit,  because  there  is   no   reason   why  any  of  the  differential 


VAN  DEB  WAALS'  EQUATION  247 

coefficients  in  (31)  should  be  zero  or  infinite.      Thence,  from  the 
Clapeyron-Clausius  equation  : 

dp_  S2  —  Si 


p_ 
dT  ~ 


ra  -  n 

we  see  that  the  vapour-pressure  curve  has  a  finite  limiting  gradient 
at  the  critical  point.  This  has  been  verified  experimentally  by 
Cailletet  and  Colardeau  (1892)  who  find, 

for  water  :  Lim  ^  =  2  '22  (p  in  atm.) 

carbon-dioxide  :  Lim  -xL  =  1'60. 
a  I 

In  the  system  considered  the  suffixes  1  and  2  refer  to  liquid  and 
vapour  respectively,  and  in  this  case  it  is  known  from  experiment 
that: 

S2  >  Si,  i.e.,  L  is  positive  i  0 

and  r2>n  I    ' 

and  hence  : 

Lim  ^  is  positive  at  the  critical  point,  and  the  vapour-pressure 

curve  slopes  constantly  from  left  to  right  upwards  to  the  critical 
point.  This  has  been  verified  by  experimenters  who  have 
followed  the  curves  as  far  as  the  critical  point. 

The  plait  point  is  an  ordinary  point  on  the  connodal  curve,  and 
hence  it  is  immediately  evident  that  the  specific  volume  and 
entropy  in  the  critical  state  are  intermediate  between  those  of 
adjectent  liquid  and  vapour  phases. 

From  (32)  we  see  that  the  fractions  in  (31)  are  all  positive,  and 
since  the  conditions  of  stability  at  points  1  and  2  require  that  : 

82U          82U 


8S  8S8r 

82U          8^U 
8S8t:          8r2 


'  ^S2 
it  follows  that,  at  the  plait  point : 


..-•     (33) 
Equations  (3),  (9)  and  (10)  give,  for  the  changes  in  tempera- 
ture and  pressure  for  motion  along  the  connodal  curve : 

/82U\  .   /82U 


248  THEEMODYNAMICS 


and  hence,  from  (31),  the  right-hand  members  of  (34)  and  (35) 
are  zero  at  the  plait  point.     Hence  at  that  point  : 

.        .        .     (86) 


-a         =  0.         ,  .     (38) 

f  =  0,^=0  .        .        .        .     (89) 
dri-  c?r2 

If  we  take  rectangular  axes,  and  put 
x  =  S,  y  =  T   x 


x  =  r,  y  =  p    > 

respectively,  we  see  that  the  two  curves  in  these  planes, 
representing  the  two  states,  meet  at  a  point  corresponding  with 
the  critical  point,  and  have  a  common  tangent  parallel  to  the 
axis  of  x. 

Further,  since  values  of  x  for  the  critical  point  are  intermediate 
between  those  on  the  two  curves,  it  follows  that  the  critical 
temperature  and  the  critical  pressure  are  maximum  or  minimum 
values,  and  we  shall  assume  they  are  the  former.  The 
inequalities  :  v%  >  i~i,  8-2  >  Si  now  lead  to  the  following  inequali- 
ties, referring  to  the  immediate  vicinity  of  the  critical  point  : 


>*<0.         .        -         .     (48) 

Now  L  =  T(S2  —  Si)    .        .        ,       ;.     (44) 

and  for  motion  along  the  connodal  curve  we  have  : 
L  =  (Sa  -  SiMT  +  TO/Sa  -  rfSi) 


VAN  DEE  WAALS'  EQUATION 

;2    rfSi 


249 


.     (45) 

From  conditions  (36)  and  (40)  we  see  that  as  the  critical  state  is 
approached,  —^  approaches  the  limit  —  oo .  If  therefore  L  is 

plotted  against  T,  the  curve  bends  round  and  turns  its  concave 
side  towards  the  T  axis,  meeting  that  axis  normally  at  the  critical 
temperature.  This  has  been  verified  experimentally  by 
Mathias  (§  92). 

Again,  since  A  =  Xra  —  *"i)          •         •         •         (46) 

it  can  easily  be  shown  that  ^-  and  ^—  tend  to  —  oc  in  the  critical 

state. 

Let  us  suppose  that  T  and  p  are  always  given  by  equation  (3), 
even  on  parts  of  the  primitive  surface  which  represent  essentially 
unstable  states.  Then  for  motion  from  one  point  on  that  surface 
to  another  point  we  have  : 

dT  =  ^ 


aau 


+  mvdl' 

82U  , 


The  solutions  of  (47)  and  (48)  are  : 


(48) 


•     (49) 


as2 


where  A  has  the  value  defined  in  (25). 
For  motion  along  an  isopiestic  :  dp  =  0, 


.     (50) 


.     (51) 
(52) 


250  THERMODYNAMICS 

For  motion  along  an  isotherm  :  dT  =  0, 


dSSe 


8S2 

and  since  A  is  zero  at  the  plait  point,  the  following  relations 
hold  for  the  critical  state  of  a  one  component  system  : 

\dS/  P       '  \dr  /  ~      '  \  (/S/  ^~     '  \dc/  ~ 

Again,  for  a  point  in  the  immediate  neighbourhood  of  the  plait 
point : 

and  if  we  suppose  that  A  is  positive  at  that  point,  equations  (51) 
(52),  (53),  and  (54)  give  : 


Since  (TQ)*   \j)    are  positive   for  any  homogeneous  phase 

in  stable  equilibrium,  the  first  and  fourth  inequalities  of  (56)  are 
verified  for  any  such  phase,  but  the  second  and  third  are  not 
necessarily  verified  except  in  the  immediate  vicinity  of  the  critical 
state. 

From  (51)  we  have  : 


8A  ,         A      ,    82U 
TT-  dv       A  .  d 


as2 

and  since  at  the  plait  point  A  =  0 


(57) 


8A  rfQ       A_ 

:  ?^_!  8t'  (58) 

82U 

8S2" 

But  5Q5-  dS  -f-  5-2-  dr  =  —  dp '=  0  .         .         .     (59) 


VAN  DER  WAALS'  EQUATION  251 

/.  on  elimination  of  dv  from  (58)  and  (59)  we  have  : 

8A  _8_A 

88  9r 

82U  82U 

(60) 


Now,  from  (30)  we  know  that  the  determinant  is  zero,  hence  at 
the  plait  point  : 


the  second,  third,  and  fourth  relations  being  obtained  similarly 
to  the  first. 

Hence  if  we  take  the  rectangular  axes  specified  in  (A),  we  see 
that  the  curve  representing  y  as  a  function  of  x  has  a  point  of 
inflexion  at  the  value  of  x  corresponding  with  the  critical  state, 

for  equations  (36)  —  (39)  show  that  ^  =  0  at  that  point,  and  we 

d2ii 
have  just  proved  that  y^  =  0.    These  relations,  and  inequalities 

(40)  —  (43)  show  that  the  tangent  at  the  point  of  inflexion  is 
parallel  to  the  axis  of  x,  and  the  curve  everywhere  else  turns  its 
convex  side  to  the  axis  of  x. 

The  most  important  case  is  the  critical  isotherm  on  the  p,  r 
diagram.  This  has  a  point  of  inflexion  at  the  critical  point, 
there  becoming  parallel  to  the  volume  axis,  and  everywhere 
else  slopes  constantly  from  right  to  left  upwards  (Rule  of 
Sarrau,  1882). 

The  investigation  above  is  due  initially  to  Gibbs  (ticient.  Papers,  I., 
43  —  46  ;  100  —  134),  although  in  many  parts  we  have  followed  the  exposition 
of  P.  Saurel  (Journ.  Phi/s.  Chem.,  1902,  6,  474—491).  It  is  chiefly  note- 
worthy on  account  of  the  ease  with  which  it  permits  of  the  deduction,  from 
purely  thermodynamic  considerations,  of  all  the  principal  properties  of  the 
critical  point,  many  of  which  were  rediscovered  by  van  der  Waals  on  the  basis 
of  molecular  hypotheses.  A  different  treatment  is  given  by  Duhem  (Traits 
de  Mecanique  chimique,  II.,  129  —  191),  who  makes  use  of  the  thermodynamic 
potential.  Although  this  has  been  introduced  in  equation  (11)  as  the  con- 
dition for  equilibrium,  we  could  have  deduced  the  second  part  of  that 
equation  directly  from  the  properties  of  the  tangent  plane,  as  was  done  by 
Gibbs  (cf.  §  53). 


252  THERMODYNAMICS 

The  properties  of  the  triple  point  could  be  deduced  in  a  similar 
manner,  since  if  we  imagine  three  states  of  a  substance  coexist- 
ing in  equilibrium,  it  is  evident  that  they  will  be  represented  by 
three  points  on  the  primitive  surface  which  are  at  the  vertices  of 
a  plane  triangle  which  is  a  triply  tangent  plane  to  the  surface. 
If  this  plane  is  rolled  about  any  of  its  sides,  it  traces  out  three 
developable  surfaces,  bounded  by  connodal  curves,  and  represent- 
ing regions  in  which  the  three  phases  are  in  equilibrium  taken  in 
pairs.  The  parts  of  the  primitive  surface  between  these  develop- 
able surfaces  represent  stable  homogeneous  states.  The  connodal 
curves  for  liquid  and  vapour  ultimately  coalesce  in  a  plait  point, 
which  is  the  critical  point  for  the  transition  of  liquid  and  vapour, 
but  there  is  no  evidence  that  the  other  two  pairs  of  connodal  curves 
are  ultimately  coincident ;  in  fact  there  is  much  evidence  to  show 
that  with  those  of  solid  and  liquid,  this  is  not  the  case.  The 
thermodynainic  investigation  of  course  leaves  this  question  quite 
open. 

The  equations  : 


.. 

may  be  used  to  determine  the  values  of  the  critical  constants 
from  any  characteristic  equation.     Thus,  Dieterici's  equation  : 


I'+.ji 

which  is  of  the  form  : 

x*  4-  aj&  +  fa* 

has  only  three  real  roots,  and  gives : 

a 


and  -5^  =  3-75. 

Clausius'  equation  : 

RT  c 


f  -a        T(f  + 
gives : 

%=  3a  +  2/3, 


cR 


TK=  \/_       8c 

'V    27R(a  -I-  8)' 


CHAPTER  IX 

THERMOCHEMISTRY 

116.     Chemical  Reactions. 

If  2  grains  of  hydrogen  and  16  grains  of  oxygen,  mixed  in  the 
gaseous  state,  are  converted  into  18  grams  of  steam  at  constant 
temperature  and  atmospheric  pressure,  the  following  changes  of 
energy  occur  (0  =  100°,  p  =  1  atm.) : 

(i.)  An  amount  of  heat  58,000  calories  =  243  X  1010  ergs,  is 
evolved.  This  is  the  calorinietrically  measured  heat  of  reaction, 
which  in  our  notation  we  must  write  negative  for  heat  erolred  .- 
therefore  Q  =  —  243  x  1010  ergs. 

(ii.)  An  amount  of  work  6'25  X  1010  ergs  is  done  on  the  system 
by  the  contraction  under  atmospheric  pressure,  or 
external  work  =  A  =  —  6"25  X  1010  ergs. 

We  therefore  have,  for  the  increase  of  intrinsic  energy, 
AU  =  Q  —  A  =  243  X  1010  —  6-25  X  1010  ergs 
.-.    AU  =  —  236-75  X  1010  ergs. 

This  denotes  the  difference  between  the  intrinsic  energy  of  the 
sy stem  (H2  +  i02)T  =  373  and  of  the  system  (H20)r  =  s73 

p  =  I  atm.  j)  =  1  atm. 

In  connection  with  such  reactions  we  may  note : 
(i.)  If  the  process  is  conducted  adynamically,  as  is  the  case 
when  a  substance  is  burnt  in  a  Berthelot  calorimeter,  i.e.,  without 
change  of  volume  of  the  reacting  system,  then  A  =  0,  and 
Q  =  AU,  so  that  (heat  of  reaction  at  constant  volume)  =  (increase 
of  intrinsic  energy). 

(ii.)  If  the  reaction  occurs  in  a  condensed  system,  i.e.,  a  system 
composed  of  liquids  and  solids  only,  the  external  work  is  negli- 
gibly small  in  comparison  with  the  heat  of  reaction,  and  —  Q 
may  be  taken  as  equal  to  the  diminution  of  intrinsic  energy. 
Thus,  in  the  neutralisation  of  1  litre  of  normal  KOH  by  1  litre 
of  normal  HN03,  A  is  only  about  0'0035  per  cent,  of  Q. 


254  THERMODYNAMICS 

117.     Thermochemistry. 

The  importance  of  the  energy  changes  accompanying  chemical 
reactions,  although  dimly  perceived  by  the  phlogistonists,  was 
first  clearly  recognised  by  their  great  opponent  Lavoisier,  who, 
in  an  investigation  with  Laplace  (Ostwald's  Klassiker,  No.  40) 
stated  as  a  self-evident  truth  that  as  much  heat  is  required  to 
decompose  a  compound  as  is  liberated  when  the  compound  is 
produced  from  its  elements.  This  is  a  special  case  of  the  first 
law  of  thermodynamics  if  the  heat  changes  are  those  at  constant 
volume.  The  law  was  first  stated  with  reference  to  thermo- 
chemistry, by  G.  H.  Hess  in  1840  (Ostwald's  Klassiker,  No.  9). 
Hess's  Principle  of  Constant  Heat- Summation  : 

The  quantity  of  heat  evolved  in  a  chemical  reaction  is  the  same 
whether  the  change  occurs  directly,  or  in  several  stages. 

This  is  equivalent  to  the  statement  that  the  evolution  of  heat 
is  dependent  solely  on  the  initial  and  final  states,  and  independent 
of  the  intermediate  states.  In  this  form,  however,  the  principle 
is  indefinite,  because  the  evolution  of  heat  will  depend  on  the 
conditions  of  the  system  whilst  the  reaction  is  occurring.  As  a 
matter  of  fact  Hess's  principle  is  strictly  true  only  when  by 
"  quantity  of  heat  evolved  "  we  understand  that  evolved  when  the 
reaction  progresses  at  constant  volume,  or  when  it  occurs  under  a 
constant  pressure* 

For  the  former  is  the  diminution  of  the  intrinsic  energy : 

_  Q(,  =  L\  -  U2 

and  the  latter  is  the  diminution  of  the  heat  function  at  constant 
pressure : 

-  Qp  =  Wi  -  Wa 

and  both  are  dependent  only  on  the  initial   and   final   stages 
(§  25). 

The  heat  of  reaction  at  constant  pressure  is : 
Qlt  =  U2  —  Ui  +  RT(wa  —  ??i)  =   Qp  +  1-985T(»2  -  Hl)  cal. 
where  «i,  ??2  are  the  numbers  of  mols  of  gases  present  before  and 
after  the  reaction. 

The    adoption    of    Hess's    principle    "without    qualification,    as    Dtihem 
(Mecaniqne   chimiqne,   1,  50)  remarks,  is  not  legitimate,  and   the   success 
which  has  attended  the  application  of  the  principle  is  due  to  the  fact  that 
the  majority  of  systems  studied  by  its  aid  have  conformed  to  one  or  other  of 
the  necessary  conditions : 

v  =  constant ......     («) 

p  =  constant ......     (ft) 


THERMOCHEMISTRY  255 

Consequences  of  Hess 's  Law  : 

If  a  reaction  may  be  instituted  in  separate  stages,  satisfying 
condition  (a)  or  (b),  then 

qa  +  qb  +  •  •  =  Q, 
where  qp  =  heat  absorbed  in  the  p-ih  stage, 

Q  =  total  heat  of  reaction. 

This  was  experimentally  verified  by  Hess,  in  the  neutralisation 
of  sulphuric  acid  by  ammonia.  The  acid  (H2S04)  was  first 
neutralised  directly  with  aqueous  ammonia  (NH3aq.),  and  then 
acids  diluted  with  successively  increasing  quantities  of  water 
were  also  brought  to  the  state  of  neutrality.  In  every  case 
(heat  of  dilution)  +  (heat  of  neutralisation  of  dilute  acid) 
=  (heat  of  neutralisation  of  strong  acid). 

H2S04  +  2NH3aq.         595'8     Sum  =  595*8 

H2S04  +  H20          77-8  „  518-9          „       596'7 

H2S04  +  2H20     116-7  „  480-5  „       597"2 

H2S04  +  5  H20     155-6  „  446'5  „       601'8 

Definitions : 

(a)  The  quantity  of  heat  evolved  in  the  formation  of  a  mol  of 
a  compound  under  specified  conditions  is  called  the  Heat  of 
Formation. 

As  variable  conditions  may  be  mentioned  the  temperature  at 
which  the  combination  occurs,  the  pressure,  and  the  states  of 
aggregation  of  the  substances. 

The  same  quantity  of  heat  is  absorbed  if  the  compound  is 
decomposed  into  its  elements,  the  conditions  being  the  same  at  every 
part  of  the  reverse  process  as  they  were  during  the  formation ; 
hence  (heat  of  formation)  =.  —  (heat  of  decomposition). 

The  majority  of  reactions  studied  by  thermochemists  have 
been  carried  out  at  room-temperature  (18°  C.),  and  the  heats  of 
formation  are  referred  to  this  temperature  unless  otherwise 
specified. 

(6)  If  a  reaction  is  more  complex  than  a  formation  or  decom- 
position of  a  compound,  e.g.,  if  it  is  a  double  decomposition  : 

AB  +  CD  —  AC  +  BD, 

the  quantity  of  heat  evolved  during  the  complete  interaction  of 
the  stoichiometric  quantities  of  the  substances  is  called  the  Heat 
of  Reaction. 

If  the  reaction  is  incomplete,  coming  to  a  standstill  when  a 


256  THERMODYNAMICS 

fraction  x  of  the  stoichiometrically  possible  change  has  occurred, 
the  heat  evolved  is  x  X  (heat  of  reaction). 

Hess  (1840)  already  suspected  that  the  heat  of  combustion  of 
a  compound  (e.g.,  HaS,  CSa,  CO)  must  be  less  than  the  sum  of  the 
heats  of  combustion  of  its  components,  and  not  equal  to  that 
sum,  as  was  previously  supposed.  The  difference  is,  as  his 
principle  indicates,  the  heat  of  formation  of  the  compound. 

Example  (Hess,  1840) :  Heat  of  Formation  of  Carbon  Monoxide. 

(Generally  one  uses  square  brackets  to  indicate  that  the  formula- 
weight  of  the  substance,  the  symbol  of  which  they  enclose,  is 
taken  in  the  solid  state,  round  brackets  to  show  that  it  is  in 
the  gaseous  state,  and  no  brackets  when  it  is  liquid.) 
[C]  +  2(0)  =  (C02)  +  94,300  cal. 
(CO)  +     (0)  =  (C0a)  +  68,000  cal. 

By  subtraction : 

[C]  +  2  (0)  -  (CO)  -  (0)  =  26,300  cal. 
/.    [C]  +  (0)  =  (CO)  +  26,300  cal. 

The  substance  indicated  by  the  same  symbol  in  two  or  more 
equations  is  in  exactly  the  same  state  in  the  reactions  represented 
by  those  equations.  In  particular,  the  different  allotropic  modi- 
fications of  a  solid  element  (e.g.,  charcoal,  graphite,  diamond  ;  or 
yellow  and  red  phosphorus)  have  different  heats  of  combustion, 
and  the  particular  form  used  must  be  specified  in  every  case. 

Favre  and  Silbermann  (1852)  measured  the  heat  of  combustion 
of  carbon  in  nitrous  oxide,  and  found : 

[C]  +  2(N20)  =  4(N)  +  (C0a)  +  133,900  cal. 

But  [C]  +  2(0)  =  (COa)  +    96,900  cal. 

.-.   4(N)  +  2(0)  =  2(N20)  -    37,000  cal. 

or    2(N)  +  (0)  =  (N20)  -    18,500  cal. 

Reactions  which  occur  with  evolution  of  heat  are  called 
exothermic  reactions ;  those  which  give  rise  to  absorption  of  heat 
if  they  proceed  directly,  are  called  cndotliermic  reactions. 

These  names  are  due  to  Berthelot,  1879. 

The  author  has  found  the  following  method  of  applying  Hess's 
law  very  simple : 

Rule :  To  find  any  proposed  heat  of  reaction  write  down  the 
chemical  equations  of  the  component  reactions  so  that  each 
symbol  appears  equally  often  on  both  sides  of  the  sign  of  equality. 
If  the  heats  of  reaction  (with  proper  signs)  have  been  inserted, 
thejinknown  heat  of  reaction  being  denoted  by  x,  then  the  latter 


THERMOCHEMISTBY  257 

may   be   found   by   adding   both   sides  and  solving  the  simple 
equation  which  results. 

Heat  of  formation  of  hydriodic  acid. 

Let  (H)  +  [I]  =  (HI)  +  xK 

(HI)  +  aq.  =  HI  aq.  +  192'01  K 
KOH  aq.  +  HI  aq.  =  KI  aq.  +  135'67  K 

KI  aq.  +  (Cl)  =  [I]  +  KC1  aq.  +  262'09  K 
137-4  K  +  KC1  aq.  =  KOH  aq.  +  HC1  aq. 

220  K  +  (HC1)  =  (H)  +  (Cl) 
173-15  K  +  HC1  aq.  =  (HOI)  +  aq. 
By  addition  we  find 

220  +  137-4  +  173-5  =  x  +  262-09  +  135'67  +  192-01 

.-.  x  =  —  59-31  K  (K  =  Ostwald's  calorie.) 
[Note  :  "aq."  is  used  to  denote  a  large  amount  of  water.] 

118.     The  Development   of  Thermochemistry. 

After  the  work  of  Lavoisier  and  Laplace,  and  of  Hess  (1840), 
thermochemistry  was  developed  mainly  by  the  simultaneous 
labours  of  Julius  Thomsen  in  Copenhagen  (Thermochemische 
Untersuchungen,  Leipzig,  1882-6),  and  Marcellin  Berthelot  in 
Paris  (Mecaniqne  chimiqtie,  1879).  The  former  in  1853  made  the 
first  application  of  the  mechanical  theory  of  heat  to  chemistry, 
setting  out  from  the  fundamental  assumption  that  the  intrinsic 
energy  of  a  body  under  the  same  conditions  is  constant,  and 
showing  that  one  aspect  of  the  principle  of  Hess  is  a  consequence 
of  this  postulate.  To  bring  thermal  magnitudes  into  relation 
with  chemical  energy,  the  heat  evolved  in  a  reaction  was  taken 
as  the  difference  of  the  energies  of  the  substances  before  and 
after  the  reaction,  a  proposition  which  is  strictly  correct  only  if 
the  change  occurs  adynamically.  To  this  consequence  of  the 
First  Law,  Thomsen  added  a  new  "principle"  which  he  developed 
from  the  views  on  chemical  affinity  then  in  vogue.  He  assumed 
that  the  heat  evolved  in  a  reaction  was  a  measure  of  the  work 
done  by  the  "chemical  forces,"  and  so  was  a  measure  of  the 
"  chemical  affinity."  Thus,  in  the  decomposition  of  an  exothermic 
compound  a  great  expenditure  of  energy  is  necessary,  and  only 
such  processes  can  bring  about  a  decomposition  which  themselves 
develop  more  heat  than  is  absorbed  in  the  decomposition. 
Metals  such  as  zinc,  iron,  and  magnesium,  the  oxides  of  which 

T.  s 


258  THEKMODYNAMICS 

are  formed  with  evolution  of  more  heat  than  is  developed  in  the 
formation  of  water  vapour  from  the  same  amount  of  oxygen,  are 
known  to  decompose  steam,  but  if  the  heat  of  formation  of  the 
oxide  is  less  than  the  heat  of  formation  of  steam,  the  metal  (e.g., 
copper,  silver,  gold)  is  not  oxidised  by  steam.  In  this  way 
Thomsen  arrived  at  the  following : 

Hypothesis:  Every  simple  or  complex  action  "of  a  purely 
chemical  nature"  is  accompanied  by  the  evolution  of  heat. 

The  criterion  of  the  possibility  of  any  reaction  was,  on  this 
hypothesis : 

2Q,-SQ,  =  a, 

where  2Q/,  SQ(  are  the  algebraic  sums  of  the  heats  of  formation 
(heat  evolved  taken  positive)  of  the  final  products  and  initial 
substances,  respectively,  and  a  is  a  positive  magnitude. 

Similar  considerations  were  advanced  by  M.  Berthelot  (1865), 
and  this  so-called  "  Principle  of  Maximum  Work  "  found  its  way 
in  some  form  or  other,  into  all  chemical  treatises.  The  following, 
among  other,  objections  were  later  brought  against  it : 

(i.)  It  implies  that  a  reaction  can  proceed  in  one  direction  only, 
viz.,  that  in  which  heat  is  evolved,  and  therefore  that  reversible 
reactions  are  impossible.  In  this  sense  it  is  a  retreat  to  the  old 
doctrine  of  affinity  due  to  Bergmann. 

(ii.)  There  are  hundreds  of  reactions  which  occur  spontaneously 
with  absorption  of  heat ;  thus  most  hydrated  salts  dissolve  in 
water  with  absorption  of  heat. 

This  was  got  over  by  saying  that  in  such  cases  there  were 
"physical  changes  "  in  which  solid  salt  became  liquid,  as  well  as 
"  chemical  changes  "  in  which  the  salt  combined  with  the  water. 
The  absorption  of  heat  attending  the  first  change  exceeded  the 
evolution  in  the  second.  To  all  such  exceptions  it  was  thought 
sufficient  to  answer  that  they  were  not  "  of  a  purely  chemical 
nature." 

In  spite  of  the  fact  that  the  general  statement  of  this 
"  principle  "  has  been  shown  to  be  false  from  all  standpoints,  it 
must  be  admitted  that  its  enunciation  was  quite  in  harmony 
with  the  spirit  of  the  times  ;  the  great  physicists  Lord  Kelvin 
(1851)  and  Helmholtz  (1847)  had  previously  formulated  an 
identical  principle  in  connection  with  galvanic  cells.  Thomsen 
and  Berthelot  went  wrong,  not  in  their  enunciation  of  the  so- 
called  "  theorem  "  as  a  working  hypothesis,  but  rather  in  their 


THERMOCHEMISTRY  259 

persistent  and  blind  retention  of  a  dogma  which  had  been  proved 
by  their  own  work  to  be  incorrect. 

J.  Willard  Gibbs  (1876)  first  pointed  out  the  correct  principle, 
but  his  statement  was  expressed  in  terms  of  differential  equations 
quite  meaningless  to  the  chemists.  Helmholtz  (1882)  (Ostwald's 
Klassiker  No.  124)  showed  in  a  more  intelligible  way  that  the 
heat  evolved  in  a  chemical  reaction  is  not  usually  a  measure 
of  the  work  done  by  the  chemical  forces  (Arleitsicerth  der 
chemischcn  Verwandtschaftkrqfte),  and  in  some  cases  the  two 
can  be  opposite  in  sign.  In  spite  of  its  deposition  from  the  rank 
of  a  natural  law,  the  Thomsen-Berthelot  principle  holds  good  in 
too  many  cases  to  be  entirely  false ;  Nernst  (1906)  has  recently 
given  a  new  interpretation  of  it  which  will  be  considered  later. 


119.     Heat  of  Reaction  and  Temperature. 

For  the  dependence  of  the  heat  of  reaction  at  constant  volume, 
we  have  Kirchhoffs  equation  (§  58)  : 


where  Ft,,  r/  are  the  total  heat  capacities  of  the  initial  and  final 
systems  at  constant  volume  : 


If  the  reaction  occurs  at  constant  pressure  we  have  a  similar 
equation : 

Qp  =  W2  -  Wi  =  (U2  +  2>V2)  -  (Ui  +  /VO 

.-.  H"  =  ~  [u  +  pV]  !2  =  f^l  2=  r/  -  rp       .    (2) 

from  §  62  (12),  where  rp,  Tp'  are  the  total  heat  capacities  of  the 
initial  and  final  systems  at  constant  pressure : 

r,j  =  ^nGp ;  iy  =  s/j'cy. 

These  equations  enable  one  to  calculate  the  heat  of  reaction  at 
any  temperature  from  its  value  at  one  temperature 

Qr2  =  QT,  +     (F'  —  T)  dT     (v,  or  p,  const.) 


f- 

J   Ti 


In  thermochemistry  one  usually  takes  heat  evolved  as  positive ; 
we  shall  denote  this  by 

H=-Q. 

8    2 


260 


THEBMODYNAMICS 


Example. — Formation  of  steani  at  constant  volume : 

H2  +  i02  =  H20 

C<H2)  =  C,(o2;  =  4-76  +  0-002440^  Mallard   and 
CPvHao,  =  5-8    +  0-005720  /  Le  Chatelier. 

(7TT 
" 
(lu 

.-.  Hfl  =  Ho  +  1-340  —  0-0010302 
where  H0  =  heat  of  reaction  at  0°  C. 

According  to  Berthelot  H  =  57,254  cal.  when  0  =  100. 

/.  HO  =  57,110  cal.,  and  from  this  the  value  of  H  at  different 
temperatures  may  be  calculated.     The  results  are  represented 


57545 


-+2  H20 


-273     0\     200     500  1000 

FIG.  52. 


2000 


by  the  curve  in  Fig.  52,  which  shows  that  the  heat  evolved 
increases  with  rise  of  temperature  to  a  maximum-at  about  650°  C., 
and  then  decreases. 

The  calculation  may  be  extended  to  specific  heats  which  are 
quadratic  functions  of  temperature,  etc.,  and  we  may  also  replace 
the  integral  of  the  true  specific  heat  by  a  mean  specific  heat 
multiplied  by  the  difference  of  temperatures  (§  6) : 

1T2 

cdT  =  c(Ta  —  Ti). 

T 

If  solids  or  liquids  participate  in  the  reaction,  their  molecular 
heats  are  of  course  included  in  forming  the  total  heat 
capacity  F. 


THERMOCHEMISTRY 


261 


Example.— 3¥e  +  4H20  =  Fe304  +  4H-2 
F  -  P  =  3CFe  +  4C,H,o  -  CF^o,  ~  4C,Hs 

=  3x6-4  +  4   X   (5-8  +  OO05720)  —  35-2  — 

4(4-76  +  0-002440) 
=  -  11-92  +  0-01312(9 
.-.  Ho  =  Ho  —11-920  +  0-0065602 
where  Ho  =  36'4  Cal.    (1  Cal.  =  103  cal.) 

The  curve  in  Fig.  53  exhibits  a  minimum  at  909°  C.     The 


2-: 


O  100 


500  1,000 

FIG.  33. 


1,600 


sense  of  the  change  of  H  with  0  is  therefore  opposite  in  the  two 
examples  considered. 

If  the   Neumann-Joule  rule   (§9)  were  true,   the  heats  of 
reaction  of  solids  and  liquids  would  be  independent  of  temperature. 


CHAPTER  X 

GAS    MIXTURES 

120.     Solutions. 

After  the  fundamental  conception  of  a  chemical  compound,  as 
distinguished  from  a  mixture,  had  been  crystallised  out  from  the 
prevailing  obscurity  by  the  classical  researches  of  Proust,  it  was 
natural  that  the  attention  of  chemists  should  have  been  chiefly 
engrossed  in  the  study  of  substances  formed  on  the  plan  of 
definite  and  multiple  proportions,  and  that  solutions,  alloys, 
amalgams,  and  glasses,  which  had  been  held  up  as  examples 
of  combination  in  indefinite  proportions  by  Berthollet,  were 
relegated  to  a  somewhat  insignificant  position.  Later  on,  when 
attention  to  these  technically  very  important  bodies  was  revived, 
a  need  was  felt  for  a  general  name  to  distinguish  such  homo- 
geneous masses  of  variable  composition  from  heterogeneous  bodies, 
or  "  mechanical  mixtures."  J.  W.  Gibbs  proposed  that  all  homo- 
geneous bodies  which  differed  in  composition  and  physical  state 
should  be  called  different  phases ;  bodies  differing  only  in  quantity 
and  form,  on  the  contrary,  would  be  different  examples  of  the 
same  phase. 

If  a  homogeneous  phase  is  formed  from  two  or  more  com- 
ponents it  is  called  a  solution  ;  it  may  be  gaseous,  liquid,  or 
solid. 

If  the  solubility  of  either  component  in  the  other  is  unlimited 
("  free  miscibility,"  as  with  alcohol  and  water),  there  may  be  an 
infinite  number  of  solutions,  lying  between  the  two  pure  sub- 
stances as  limiting  cases.  The  solubility  may  be  limited  in  one 
or  both  directions.  Thus,  water  and  salt  form  a  series  of 
solutions  extending  indefinitely  towards  pure  water  as  one  limit, 
but  bounded  by  saturated  salt  solution  as  the  other  limit ;  water 
and  ether  form  a  continuous  series  of  solutions  bounded  on  one 
side  by  a  saturated  solution  of  ether  in  water,  and  on  the  other 
side  by  a  saturated  solution  of  water  in  ether.  In  the  region  of 
continuous  miscibility  all  the  properties  of  the  solution  vary 


GAS  MIXTURES  263 

in  a  continuous  manner  with  the  composition ;  examples  of  such 
properties  are :  density,  refractive  index,  light  absorption  (colour ), 
specific  heat,  coefficient  of  expansion,  compressibility,  etc.  In 
some  cases  the  value  of  the  property  for  the  solution  can  l>e 
calculated  additively,  from  the  values  of  the  pure  components, 
and  the  relative  amounts  of  these  present  in  the  solution  : 

(l>i  +  7*2  +  Pa  +  •  •  •)  T  =  7iai  +  7-2«2  +  7a«3  -f  •  •  • 
where  pi,  p*  are  the  amounts  of  the  components ;    a\,  a-2  the 
specific  values  of  the  property.     This  is  called  the  Mixture  Rule, 
and  solutions  which  satisfy  it  are  often  called  "  mixtures." 

121.  Concentration. 

It  is  convenient  to  have  some  uniform  method  of  representing 
the  composition  of  a  solution,  and  for  this  purpose  use  is  made 
of  the  so-called  concentration  of  a  component,  which  may  be 
referred  to  the  total  mass,  or  to  the  total  volume,  or  to  the 
molecular  constitution. 

Let  u~i,  ic-2,  ...  be  the  weights  of  the  various  components ; 
MI,  n-2,  ...  the   numbers   of   uiols  of   the  various   com- 
ponents ; 
V  the  total  volume,  TV  the  total  weight,  and  X  the  total 

number  of  mols. 
Then: 

(ci)  =  — ?,  .  .  .  are  the  weight  concentration*, 
[ci\  =  =i,  ...  „  volume  concentrations, 
Ci  =  ^, numerical  concentration*, 

$i     =  -i,    .  .  -.         „        rolu  metric  molecular  concentration*. 

If  one  component  is  present  in  such  small  amount  that  its 
ic,  n,  or  r,  may  be  omitted  in  forming  the  sums  W,  N,  or  V,  the 
solution  is  called  dilute  with  respect  to  that  component. 

122.  Mixtures  of  Ideal  Gases. 

The  phenomena  accompanying  the  admixture  of  gases  were 
first  accurately  studied  experimentally  by  John  Dalton  (1801), 
who  arrived  at  the  following  conclusions : 

(1)  If  two  or  more  gases  are   placed   in    contact,  either  by 


264  THERMODYNAMICS 

stratifying  in  layers,  or  by  connecting  together  the  vessels  con- 
taining the  gases,  then  after  the  lapse  of  a  certain  time,  which 
depends  to  some  extent  on  the  nature  of  the  gases,  a  homogeneous 
mixture  is  produced  which  never  spontaneously  separates  into 
its  components.  This  spontaneous  admixture  of  gases,  called 
diffusion,  was  thus  established  as  a  real  property  of  gaseous 
substances,  in  opposition  to  the  views  of  Priestley  and  others, 
that  if  the  gases  were  placed  together  in  layers,  the  lighter  ones 
being  uppermost,  they  would  remain  separate,  like  oil  and  water, 
unless  mechanically  agitated. 

(2)  If  different  gases  are  mixed  by  diffusion,  at  the  same  tempera- 
ture and  pressure,  then  (provided  no  chemical  action  occurs) : 

(a)  The  volume  of  the  mixture  is  the  sum  of  the  volumes  of 
the  constituent  gases ; 

(6)  The  pressure  remains  unchanged  throughout  the  process. 

Definition  of  an  Ideal  Gas  Mixture.— It  a  mixture  formed  of  the 
masses  MI,  ?n2,  .  .  >nn  of  the  ideal  gases  GI,  G2,  .  .  G?i  has  a  free 
energy  equal  to  the  sum  of  the  free  energies  of  these  masses  of 
the  separate  gases,  at  the  same  temperature  and  each  occupying 
a  volume  equal  to  the  total  volume  V  of  the  mixture,  it  is  called 
an  ideal  gas  mixture. 

The  justification  of  this  definition  will  be  considered  later;  at 
present  we  shall  show  that  it  leads  to  consequences  in  agreement 
with  experience. 

The  equations  are  greatly  simplified  if  we  refer  everything  to 
molecular  quantities.  Let : 

nil,  viz,   •  •  w>i  be  the  masses, 

i'i,    r2,    .  .  rh  the  specific  volumes, 

MI,  M2,  .  .  M,,  the  molecular  weights, 

V'l*  ^2,    •  •  $h  ^e  free  energies  per  unit  mass, 

C^,  Ci2),  .  .  C(r\  the  molecular  heats  at  constant  volume, 

MI  =  /MI/MI,  na  =  wia/Ma,  .  .  n,  =  w,-/M;,  the  number  of  mols, 
of  the  gases  GI,  G2,  •  •  G,-,  in  the  mixture. 

The  free  energy  $,  of  unit  mass  of  the  i-th  gas  is  (§  79) : 


-  felJP  -  T  fe 


where  <7j'(T)  =  u0  —  Ts0  -f  |  <-  v  ±  —  j.  | 

and  the  free  energy  of  the  mass  m-t  is,  since  m,  =  w,M; : 


GAS  MIXTURES  265 


But  if  V  is  the  total  volume  : 

r<  =  V/ra,  =  V/n,.M, 


where  </,<T)  =  M#,'(T)  +  RTJwM,  is  also  a  function  of  tempera- 
ture, and  is  constant  for  a  mol  of  a  specified  gas  at  a  particular 
temperature. 

If  the  specific  heats  are  all  independent  of  temperature  : 

M.-M^,-  =  n,<Uif'  -  TSi°)  +  n.-TXCS"'  -  UOf^.vf} 

.-.  ^/(T)  =  U0"  -  TSi"  +  T(Ci!'  -  /»Tc<r"). 

If  the  specific  heats  are  not  constant  we  can  use  equations  (12) 
of  §  79. 

From  the  definition  of  an  ideal  gas  mixture,  we  shall  have  for  the 
free  energy  of  the  mixture  of  i  gases  in  the  volume  V  the  expression  : 


.     (5) 

But  if  Pi  is  the  pressure  which  would  be  exerted  by  the  t'-th 
gas  in  solitary  confinement  in  the  volume  V,  we  have : 


.-.*=-  2»,-  [RT/W  ?  +  RT/wR  -  r/,<T)]  .     (7) 

If  p  is  the  total  pressure  of  the  mixture,  we  have  : 


/.  by  differentiating  (5)  partially,  with  respect  to  V  : 

,  =  X^.*»    ....    (9) 

so  that  t/ie  fo/oZ  pressure  of  an  ideal  gas  mixture  is  the  sum  of  the 
pressures  which  each  gas  would  exert  if  separately  confined  in  the 
space  occupied  />//  the  mixture,  and  with  a  free  enerni/  equal  to 
that  which  it  possesses  in  the  mixture. 

Pi  is  called  the  partial  pressure  of  the  i-th  gas  in  the  mixture, 
and  hence  (9)  states  that  the  total  pressure  is  the  sum  of  the 
partial  pressures. 

This  is  nothing  else  than  the  well-known  Law  of  Mixed  Gases, 
or  Law  of  Partial  Pressures,  discovered  experimentally  by  John 
Dal  ton  in  1801,  and  expressed  by  him  in  the  somewhat  vague 


266  THERMODYNAMICS 

statement   that   "  one   gas   acts   as   vacuum   towards   another  " 
(Manchester  Memoirs,  5,  550,  1802). 

Let  us  put  M  =  ^MH 


where  M  is  called,  for  convenience,  the  mean  molecular  ireinlit  of 
the  gas  mixture.  In  assigning  a  molecular  weight  to  a  mixture 
we  merely  state  what  weight  of  that  mixture  occupies,  under 
normal  conditions,  the  same  volume  as  32  grams  of  oxygen  ; 
a  mixture  of  course  has  no  "  molecular  weight  "  at  all  in  a 
purely  chemical  sense. 

If  we  put  n  =  iii  -\-  nz  +  •  •  +  ni 

then  ?*M  =  n^  +  »2M2  +  .  .  +  n^     .         .         .     (11) 

which  is  simply  an  expression  for  the  law  of  conservation  of  mass. 

The  density  p  of  the  mixture  is  obviously  equal  to  -^-  and  the 
density  pt  of  the  i-ih  constituent  when  in  solitary  confinement  in 
the  same  volume  is  -J^T:I,  hence,  from  (9)  and  (11)  : 

P  =  Pi  +  P*  +  •  •  +Pi  =  2p,  (12) 


But  if  we  put  pf  =  —,  where  <•/  is  the  specific  volume  of  the  i-th 

component  in  the  mixture  (not  to  be  confused  with  /•,)  : 
1        1     ,     1  1 


or  v  =  T-  —  p      .         .         .     (13) 

Z+X+-+Z 

which  gives  the  specific  volume  of  the  mixture. 

Corollary.  —  A  gas  mixture  obeys  the  general  gas  law. 
The  free  energy  of  n:  mols  of  a  single  gas  is  : 

nMrti  =  -  »4BTf»—  4-  »,-r/{(T). 

The  free  energy  of  w  mols  of  a  gas  mixture  of  mean  molecular 
weight  M  is  : 


.        .     (14) 
where         G  =       i  .  . 


GAS  MIXTURES 


267 


It  therefore  follows  that  free  energy  of  an  ideal  gas  mixture  is 
a  function  of  the  same  form  as  that  of  a  simple  gas.  Heiice,  in 
virtue  of  Massieu's  theorem,  an  ideal  gas  mixture  behaves  ther- 
mally and  mechanically  exactly  like  a  simple  gas. 

In  particular,  if  Ct>,  Cp  are  the  molecular  heats  at  constant 
pressure,  and  at  constant  volume,  respectively,  of  the  mixture, 
and  Cp1',  C;?',  those  of  the  first,  CJ?,  Cf,  those  of  the  second, 
component,  and  so  on, 


(16) 

=  C,  +  R 

and  it  is  easy  to  show  from  this  and  the  preceding  equations 
that  : 

c  = 


"i  +  »«  + 


III!  + 

so  that  the  specific  heats  of  the  mixture  are  calculable  from  those 
of  the  constituents,  at  an  assigned  temperature,  by  the  mixture 
rule. 

Now  suppose  we  have  the  gases  GI,  G-2,  .  .  G,-,  in  the  amounts 
specified  at  the  beginning  of  this  section,  all  at  the  same  tempera- 
ture T,  separately  confined  in  vessels  of  volumes  Vi,  V->,  .  .  V  , 
respectively.  If  we  add  together  the  free  energies  of  the  separate 
gases,  the  sum  *o  may  be  called  the  free  energy  of  the  unmixed 
gases.  Thus  : 


.        .     (18) 

Now  let  all  the  vessels  be  put  in  communication,  and  let  the 
gases  mix  so  as  to  form  a  homogeneous  gas  mixture  of  volume  : 

.        .     (19) 


268  THERMODYNAMICS 

The  free  energy  of  the  mixed  gases,  *,  has  already  been  calcu- 
lated ;  from  (3)  it  is  : 

— 

+  n 


.        .     (20) 

Thence,  the  diminution  of  free  energy  incurred  by  the  mixing 
of  the  gases  is  : 

*„  -  *  =  ET 


(21) 


which  is  evidently  positive.  Hence  diffusion  is  a  genuine  spon- 
taneous phenomenon  exhibited  l>y  gases,  which  result  was  experi- 
mentally established  by  Dalton. 

Further,   if  Uo,  U  are    the    total   intrinsic   energies   of   the 
unmixed  and  mixed  gases,  respectively  then  (§  58)  : 


=  (*0  -  >IO  -  T          p      .         .         .     (22) 

But  we  see  by  differentiating  (16)  and  multiplying  the 
result  by  T,  that  the  expression  on  the  right  of  (22)  vanishes,  so 
that  : 

Uo-U  =  0) 
or  U  =  U0     ) 

Corollary.  —  If  different  ideal  gases  mix  by  diffusion  so  that  the 
total  volume  of  the  mixture  is  equal  to  the  sum  of  the  volumes  of 
the  constituents,  there  is  no  evolution  or  absorption  of  heat.  This 
result,  which  may  be  regarded  as  an  extension  of  the  theorem  of 
Joule,  was  also  experimentally  discovered  by  Dalton. 

For  the  purpose  of  throwing  the  equations  into  convenient 
forms  as  required,  we  may  deduce  a  few  simple  relations. 


GAS  MIXTURES  269 

We  have  Vi/Vg  =  wi/«a,  by  Avogadro's  theorem,  hence  in  virtue 
of  a  well-known  algebraical  theorem  : 

Vi  +  V2  +  ^  +  V,       /n  +  na  +  .  .  +  MJ 
"Vi  ttl 

or  generally,  SV,-  _  S», 

V;     ~~     »; 

If  Pi,  lh,  •  •  J^;  are  the  partial,  and  p  the  total,  pressures  : 

*  =  v,  +  v,+  ..  +  v,»    "=v,  +  v,+i.  .  +v,^ 

and  generally  ^,.  =      '  ^>    .         .         .         .     (25) 

If  Ci,  c2,  ....  c;  are  the  numerical  concentrations  : 

Cj  =  W../S/I,-  =  Vf/SV,-.  .         .         .         .     (26) 
It  is  also  evident  that  the  numerical  concentrations  are  pro- 
portional to  the  partial  pressures  : 

Pi  =  ^-P  =  cip    ....     (27) 

If  £ii  &i  '  •  •  •  &  are  the  volumetric  molecular  concentrations, 
6  =  n,/V  =  w,/2Vi 


But  V  =  »— 


.     (28) 


Equation  (23)  shows  that  the  energy  of  the  mixture  is  the  sum 
of  the  energies  of  the  constituents,  and  is  independent  of  the 
volume.  Thus  if  : 


I  C 

J    0 


rfT          .        .         .     (29) 
is  the  energy  of  a  mol  of  the  i'-th  gas, 

nU  =  «Ma  =  S/iiUj  =  SwjUi4'  +  Sn{    Cy»dT       .        .     (30) 


270  THERMODYNAMICS 

or,  if  C[f  is  assumed  independent  of  temperature  : 

«U  =  2,-H,-[Uo  +  HiC'j.'T]  .  -  v  .  (31) 
where  »,-,  M;,  C'r",  M,  have  the  usual  significance,  and  n~M.u  is 
the  whole  energy  of  the  mixture,  U  its  energy  per  mol,  and  W 
is  the  arbitrary  term  in  the  expression  for  the  energy  of  the 
i-th  gas. 

For  the  free  energy  of  the  gas  mixture  we  have  : 

wM^  =  —  2»i  [~BTJn  J   •-  gi(T)~\ 

^>  -  L  f]         .         .     (32) 
-  Kin  *j  +  Btocf]   .     (33) 

+  Binfi]    .         .         .     (34) 

from  (27),  and  (28). 

If  the  specific  heats  are  independent  of  temperature,  we  obtain 
//;(T)  from  (4),  and  hence  : 

nlVty  =  T2w;  fcii'Cl  -  ZwT)  -  B/u  -  -  M^S^  -  B?w  ^  +  M^ 
L  Pi  M,  1- 

-  iwT)  -  B/w  -  -  M{Sy  -  R  fo*      - 
P 


.     (35) 
BiwM,.    +    B/»f(- 

.    (36) 

The  potential  is  obtained  simply  by  adding  2w,-BT  to  the  expres- 
sions for  the  free  energy  (cf.  §  79)  : 

•     (37) 


s  , 

=  T2«,  [^  -  Bfot  ^  +  R(l  +  toe,)]       .        .     (88) 
If  the  specific  heats  are  independent  of  temperature  : 

-  B/-H  -^    +  R  —  8^  -  B/M  ~ 

TTl^-i 

4--  -     (39) 


GAS  MIXTURES  271 

=  T2nf  [cr"'(l  -  /«T)  -  R/ii  -^  +  B  -  SJf' 


•  (40) 

The  entropy  is  obtained  from  the  free  energy  by  means  of  the 
equation  S  —  —  ^~: 


/I       -  R(l  -f-  /we,-)  - 
If  the  specific  heats  are  assumed  to  be  constant  : 


/*M*  =  2w,-     c^/wT  +  R/H       +  S0"  +  R/»        -  fact        (42) 

Examples.  —  il)  Prove  that  the  entropy  of  an  ideal  gas  mixture  is  the  sum 
of  the  eutropies  of  the  components  at  the  same  temperature,  each  occupying 
the  whole  volume  of  the  mixture. 

(2)  Show  that  the  potential  of  an  ideal  gas  mixture  is  the  sum  of  the 
potentials  of  the  components  at  the  same  temperature,  each  occupying  the 
whole  volume  of  the  mixture. 

(This  result  is  very  important.) 

(3)  Show  that  the  diminution  of  free  energy  on  mixing  «i.  n.2,  .  .  niols  of 
the  gases  GI,  G*,  .  .  so  that  the  total  volume  remains  constant  is  : 

V0  —  *i  =  BT2H.ZH  _^   =  BTStUn—  =  -  KTSn./nc 
»i  Pi 

=  -  BTSi^n*.  +  2ft7n  JL- 

(4)  For  two  gases,  with  HI  =  »a  =  1,  the  maximum  loss  is  incurred  with 
equal  volumes,  and  is  BTf»2. 


123.     Verification. 

All  the  expressions  for^o  —  *i  evidently  represent  the  di$si2>a- 
tion  of  energy  which  occurs  when  the  gases  are  allowed  to  mix  by 
diffusion  in  the  specified  manner.  It  follows  from  the  principle 
of  dissipation  of  energy  that  work  will  have  to  be  spent  in  sepa- 
rating the  mixture  into  its  constituents,  and,  conversely,  work 
should  be  obtained  if  the  gases  are  allowed  to  mix  in  a  suitable 
manner.  The  first  quantity  of  work  will  be  a  minimum,  the 
latter  a  maximum,  and  both  equal  and  opposite,  when  the  pro- 
cesses are  conducted  reversibly. 

The  definition  of  an  Ideal  Gas  Mixture  given  in  §  122,  although 
it  leads  to  results  in  entire  accord  with  those  established  bv 


272 


THERMODYNAMICS 


experiment,  cannot  be  regarded  as  entirely  justified  unless  we 
can  show,  by  some  physical  method,  that  the  free  energy  of  a  gas 
mixture  is  less  than  the  sum  of  the  free  energies  of  its  components 
in  the  free  state  at  the  same  temperature  and^  pressure  by  the 
amount  of  work  which  is  obtained  in  producing  the  mixture 
isothermally  and  reversibly  from  its  components. 

This  has  been  calculated  from  the  definition,  and  the  result  is 
contained  in  (21),  §  122. 

The  solution  of  the  problem  depends  solely  on  the  possibility 
of  finding  a  process  by  which  a  gas  mixture  can  be  formed 
reversibly  from  its  components,  or  the  mixture  separated  reversibly 
into  the  latter. 

Passing  over  such  special  methods  as  separation  by  differences 


(I) 


(2) 


A  B 
FIG.  54. 

of  solubility  in  a  liquid,  or  by  different  condensation  tempera- 
tures, we  come  to  a  general  method  of  separation  of  gaseous 
mixtures  introduced  into  the  theory  of  the  subject  by  Rayleigh 
(1875),  and  Boltzmann  (1878).  This  method  has  the  great 
advantage  that,  by  its  aid,  the  separation  may  be  effected  isothcr- 
mally  and  reversibly.  It  depends  on  the  existence  of  substances 
exhibiting  a  selective  permeability  to  gases;  such  as  palladium, 
platinum,  and  iron  at  high  temperatures,  which  are  freely 
permeated  by  hydrogen,  but  not  by  nitrogen. 

It  is  therefore  legitimate  to  postulate,  for  the  purposes  of 
thermodynamic  reasoning,  ideal  septa  each  of  which  is  permeable 
to  one  gas  but  quite  impervious  to  all  others.  Such  septa  are 
called  semipermeable  septa. 

We  now  suppose  that  we  have  (Fig.  54)  an  impervious  cylinder 
fitted  with  two  semipermeable  pistons  A  and  B,  connected  to  some 
outside  sources  or  receivers  of  work  by  rigid  piston  rods,  passing 
gas-tight  through  the  cylinder  ends. 

Let  the  pistons  be  first  placed  in  contact,  and  let  A  be  freely 


GAS  MIXTURES  273 

permeable  to  an  ideal  gas  [1],  contained  in  the  space  (l)of  volume 
r  i  ;  let  B  be  freely  permeable  to  an  ideal  gas  [2]  ,  contained  in  the 
space  (2)  of  volume  r2.  Each  piston  is  impervious  to  the  gas 
which  penetrates  the  other. 

Then  it  is  evident  that  no  pressure  is  exerted  on  either  piston 
by  the  gas  which  freely  permeates  it,  and  the  pressure  to  which  a 
piston  is  exposed  is  therefore  always  that  exerted  upon  it  by  the 
gas  to  which  it  is  impervious. 

The  pistons  therefore  tend  to  separate,  A  under  the  pressure 
exerted  by  the  gas  [2],  B  under  the  pressure  exerted  by  the  gas 
[1],  and  motion  ceases  only  when  the  pistons  are  in  contact 
with  the  cylinder  ends,  and  the  gases  are  uniformly  mixed. 

The  mixing  can  be  performed  isothernially  and  reversibly  by 
sinking  the  apparatus  in  a  constant  temperature  bath,  and 
opposing  the  expansive  forces  of  the  gases  by  forces  differing 
only  infinitesimally  from  them,  and  applied  to  the  piston  rods. 

The  slightest  increase  of  applied  force  will  reverse  the  direction 
of  the  process,  and  the  mixture  is  separated  reversibly  into  its 
constituents. 

If  there  are  HI  inols  of  gas  [1]  and  n-2  mols  of  gas  ~2~,  the 
work  done  during  the  mixing  is  equal  to  the  sum  of  the  amounts 
of  work  done  by  each  gas  hi  expanding  from  its  initial  volume  to 
the  final  volume  of  the  mixture,  for,  as  is  evident  from  the 
conditions,  each  gas  performs  work  as  if  the  other  were  absent. 

H!  mols  of  [1~  expand  isothernially  and  reversibly  from  a  volume 
t~i  to  a  volume  (i'i  -\-  r2)  : 

.-.  A,  =  MlRT7it  <!±±V. 

*'i 

its  mols  of  [2]  expand  isothermally  and  reversibly  from  a  volume 
r-2  to  a  volume  (rx  +  r.2)  : 


'2 

ii^±^  +  i^ii^-±^l    .    (1) 

'"i  *'a     J 

This  expression  (which  can  easily  be  generalised)  for  the  maxi- 

mum work  agrees  with  that  obtained  above  for  the  diminution  of 

free  energy,  if  there  are  two  components,  and  the  justification  of 

the  definition  is  established. 

Further,  since  there  is  no  change  of  internal  energy  on  mixing 

the  gases,  the  expression  (1)  will  also  represent  the  heat  absorbed 


274  THERMODYNAMICS 

in  the  isothermal  and  reversible  process,  and  if  this  is  divided  by 
the  temperature,  we  obtain  directly  the  increase  of  entropy: 

A  S  =  R     niln  —     — -  +  n^ln  —     — -      .          .          .     (2) 

V\  1'2        J 

124.     Gibbs's   Paradox. 

The  preceding  calculation  of  the  work  done,  and  heat  absorbed 
by  the  isothermal  and  reversible  mixing  of  two  gases  does  not 
depend  on  the  physical  nature  of  the  gases,  and  is  therefore  applic- 
able to  the  mixing  of  such  gases  as  nitrogen  with  carbon  monoxide, 
although  both  these  (having  equal  densities)  are  physically 
identical.  If,  however,  the  gases  are  chemically  identical,  no  work 
is  done,  for  then  the  same  material  must  be  used  for  both  pistons, 
and  neither  gas  exerts  pressure  on  a  piston.  The  necessary  con- 
dition that  work  can  be  obtained  in  the  reversible  mixing  of  gases 
is  that  the  gases  shall  be  chemically  different. 

The  mixing  of  gases  was,  after  Rayleigh,  discussed  by  Gibbs 
(1876),  who  obtained  the  general  equation  (21),  §  122,  from  which 
the  work  obtained  by  the  isothermal  and  reversible  mixing  of  gases 
can  be  calculated.  There  appeared,  however,  no  reason  why  this 
equation  should  not  apply  equally  well  to  two  portions  of  the  same 
gas,  and  hence  arose  the  so-called  "Gibbs's  Paradox."  The 
verification  depending  on  the  use  of  semipermeable  septa  con- 
tains in  itself  an  explanation  of  the  supposed  difficulty.  The 
kinetic  theory  of  gases  does  not  throw  any  light  on  the  subject, 
because  two  gases  of  equal  density  are,  from  the  standpoint  of 
statistical  dynamics,  identical.  These  may,  however,  behave 
quite  differently  towards  a  third  gas,  according  as  this  is,  or  is  not, 
chemically  identical  with  one  of  the  gases,  although  in  no  case  is 
there  any  chemical  change.  Thus,  carbon  monoxide  and  nitrogen 
furnish  work  on  reversible  admixture,  not  so  two  portions  of 
either  gas. 

125.     Deviations  from  the  Law  of  Partial  Pressures. 

The  law  of  Dalton  is  a  limiting  law  which  is  never  followed 
perfectly  strictly,  although  the  deviations  with  the  permanent 
gases  under  moderate  pressures  are  very  small.  Andrews  (1876) 
found  that  there  was  an  expansion  when  strongly  compressed 
nitrogen  and  carbon  dioxide  were  mixed.  Braun  (1888)  observed 


GAS   MIXTURES  275 

that  there  is  a  decrease  of  pressure,  below  the  sum  of  the  partial 
pressures,  when  sulphur  dioxide  is  mixed  with  carbon  dioxide  or 
hydrogen  ;  whilst  if  hydrogen  is  mixed  with  carbon  dioxide,  air, 
or  nitrogen,  there  is  an  increase  of  pressure. 

In  general,  according  to  Galitzine  (1890),  the  sum  of  the  partial 
pressures  is  a  little  greater  than  the  total  pressure,  at  low  tem- 
peratures. With  rise  of  temperature  the  deviation  passes  through 
a  maximum,  and  then  changes  sign. 

Further  information  will  be  found  in  Kuenen  :  Verdampfimg 
und  Vcrflustigung  von  Grmisclu'tt,  99  —  106,  which  contains  a 
bibliography  ;  cf.  also  Young:  Stoichiometry. 

126.     Solubility   of   Gases    in    Liquids;   Laws    of    Henry   and 
Dalton. 

A  gas  when  brought  in  contact  with  a  liquid  dissolves  to  a 
greater  or  less  extent  according  to  the  particular  chemical  natures 
of  the  two  substances,  and  the  solution  so  formed  comes  into 
equilibrium  with  the  excess  of  gas  standing  above  it.  This 
equilibrium  is  characterised  by  a  very  simple  law,  discovered  by 
the  Manchester  chemist  William  Henry  (1803),  and  called  after 
him: 

Henry's  Law  :  At  a  given  temperature  the  amount  of  gas 
dissolved  in  a  liquid  solution  at  equilibrium  is  proportional  to 
the  pressure  in  the  gas  space.  . 

If  Hi  is  the  mass  of  gas  dissolved  in  a  given  volume  of  a  liquid 
under  unit  pressure  at  a  given  temperature,  /*,,  the  mass  dissolved 
under  a  pressure  p,  then,  by  Henry's  law 

IS=l»-Mi      •          •          .          .         (1) 

If  pi,  pp  are  the  corresponding  densities  of  the  gas,  the  col  nines 
absorbed  are  ^  — 


fi         Hip?         PP 

since  pp  :  p\  =  p  :  1  by  Boyle's  law. 

Thus  the  volume  of  gas  dissolved  by  a  given  volume  of  liquid 
at  a  given  temperature  is  independent  of  the  pressure.  It  follows 
at  once  that  the  volumetric  concentrations  in  the  saturated 
solution  and  in  the  gas-space  are,  at  a  given  temperature,  always 
in  a  constant  ratio,  A.  This  ratio  A  is  called  the  solubility  of 
the  gas  ;  it  is  independent  of  the  pressure,  but  alters  with  the 
temperature. 

T  -2 


276  THERMODYNAMICS 

Corollary. — If  a  volume  r  of  gas  is  dissolved  by  a  volume  Y  of 
liquid  at  a  given  temperature  6, 

A*=r/V        ....        (3) 

Bunsen  (1855),  to  whom  we  owe  the  first  accurate  measure- 
ments of  the  solubilities  of  gases  in  liquids,  expressed  his  results 
in  terms  of  an  absorption  coefficient  @,  which  he  defined  as  the 
volume  of  gas,  reduced  to  0°  C.  and  76  cm.,  dissolved  by  1  c.c. 
of  the  liquid  at  any  given  temperature  under  the  same  pressure. 
If  r  c.c.  of  gas  are  dissolved  by  Y  c.c.  of  liquid  at  a  temperature 
0  and  pressure  p  cm.,  the  volume  reduced  to  normal  conditions  is 

v  iTfi/T^T — ff\'     r^ie  quantity  of  gas  dissolved  under  atm.  pressure 

is,  by  Henry's  law,  76/p  times  this,  and  lastly,  that  for  unit 
volume  of  liquid,  i.e.,  the  absorption  coefficient  at  the  tempera- 
ture 6  is 

fi   -  r  V  7A-  v  Xe 

~  V  '  76(1  +  a&)  '  p  ~  (I  +  a<9)V  ~  1  +  ad'   ' 

Pressure  Change  during  Solution  : 

A  mol  of  gas,  having  a  solubility  A  in  a  given  liquid,  is  con- 
tained with  a  specified  volume  of  the  liquid  in  a  cylinder  under  a 
piston.  Let  r,V  be  the  initial  volume  of  the  gas,  and  the  volume 
of  the  liquid,  respectively,  and  let  p$  be  the  pressure  of  the  gas 
when  isolated  from  the  liquid.  To  find  the  pressure  p  when  the 
volume  has  been  reduced  to  x  in  contact  with  the  liquid, 

We  have  pQv  =  RT, 

and  px  =  »RT, 

where      n  =  no.  of  mols  of  gas  left  undissolved 
1 — 11=      „         „          ,,     dissolved. 

concentration  in  liquid          /I  —  n\       n 

Now         A  =  -  — -. . —  =  I      TT— )  -r-- 

concentration  in  gas  space       \     \     /       x 

_  (1  -  n}x 


The  solubility  of  a  gas  in  a  liquid  usually  decreases  with  rise 
of  temperature  ;  those  of  the  inactive  gases  (He,  A,  etc.)  exhibit 


GAS  MIXTURES  277 

minima,  and  then  increase  (Estreicher,  Zeitschr.  pliysik.  Chem. 
31,  176,  1899;  v.  Antropow,  Proc.  Roy.  Soc.  A,  88,  474). 

Deviations  from  Henry's  law  are  exhibited  by  most  gases 
having  absorption  coefficients  greater  than  100.  In  some 
cases  the  discrepancies  vanish  at  higher  temperatures.  Thus 
Eoscoe  and  Dittmar  (1860)  found  that  ammonia  did  not  follow 
the  law  of  Henry  at  the  ordinary  temperature,  but  Sims  (1862) 
showed  that  the  deviations  from  the  law  became  less  as  the 
temperature  at  which  absorption  occurred  increased,  until  at  100° 
the  amount  of  ammonia  dissolved  by  water  was  directly  propor- 
tional to  the  pressure.  The  deviations  appear  to  be  always 
greatest  under  small  pressures,  and  to  decrease  with  increasing 
pressure,  and  therefore  with  increasing  concentration  of  the 
solution  ;  they  are  doubtless  due  to  chemical  interaction 
between  the  solvent  and  dissolved  gas. 

If  the  gas  consists  of  a  mixture  of  two  or  more  simple  gases 
(e.g.,  carbon  dioxide  and  oxygen),  which  dissolve  in  a  liquid, 
we  can  assume  as  a  first  approximation  that  the  amount  of  each 
dissolved  will  be  independent  of  the  presence  of  the  other  gases, 
and  will  be  proportional  to  its  partial  pressure  in  the  gas  mixture 
standing  in  equilibrium  over  the  solution  (J.  Dalton,  1807). 

Thus,  if  the  mixture  of  carbon  dioxide  and  oxygen  is  shaken 
up  with  water,  both  gases  will  begin  to  pass  into  solution,  and 
their  partial  pressures  in  the  gas  mixture  will  change.  When 
the  amount  of  each  in  solution  stands  in  a  definite  ratio  to  its 
partial  pressure,  there  will  be  equilibrium,  but  the  amounts 
dissolved  will  not  be  the  same  as  if  each  gas  had  been  separately 
brought  in  contact  with  the  same  quantities  of  liquid  under  a 
pressure  equal  to  its  initial  partial  pressure  in  the  mixture. 

The  law  of  Dalton  received  experimental  confirmation,  in  so 
far  as  sparingly  soluble  gases  are  concerned,  by  the  extensive 
eudiometric  researches  of  Bunsen  (1855),  who  shook  up  the  gas 
mixture  with  the  liquid  in  a  tube  ("  absorptiometer  ")  in  which 
the  pressure  and  volume  were  variable,  and  analysed  the  solution 
and  resulting  gas  mixture. 

Let  the  pressure  and  volume  of  the  mixture  before  and  after 
absorption  be  Po,Vo  and  P,V  respectively.  Then  : 


by  Dalton's  law  of  partial  pressures. 


278  THERMODYNAMICS 

We  have  also  the  equations  : 

V0  =  r0  =  i 

and  />oVo  —  P~V  ==  ^1)X     1 

.      (8) 


from  (5).     By  addition  and  substitution  from  (1)  : 
PoVo-PV 


p_  VpPo/P  -  V  -  \' 
p  - 


, 
p  -  (A  _  v;a? 

which  gives  the  concentration  in  the  gas  space  after  equilibrium 
is  attained. 

The  absolute  values  of  the  solubilities  of  gases  are  not  at  present 
calculable  from  any  general  law,  although  W.  M.  Tate  (1906) 
finds  in  the  case  of  aqueous  solutions  a  relation  with  the 
viscosities  of  the  solution  (ju.fl),  and  water  (^o),  the  critical 
temperatures  of  the  gas  (Tf/),  and  of  water  (T!(.),  and  the  absorp- 
tion coefficients  : 


(Meddd.  fran  Vetensk.  Abaci.  Nobel  hist.  1,  4,  1906,  Centralbl  , 
1908,  i.  1659.) 


CHAPTER  XI 

THE    ELEMENTARY    THEORY   OF    DILUTE    SOLUTIONS 

127.     Osmotic  Pressure. 

The  relations  of  liquids  to  seinipermeable  septa  were  observed 
by  the  French  natural  philosopher,  the  Abbe  Nollet  (1748),  who 
tied  a  piece  of  bladder  over  the  mouth  of  a  jar  containing  alcohol, 
and  placed  the  whole  in  water.  The  bladder  swelled  up,  and 
ultimately  burst  from  the  internal  pressure.  The  same  observer 
gave  a  correct  interpretation  of  the  phenomenon,  which  arises 
from  the  much  more  marked  permeability  of  the  septum  to  water 
as  compared  with  alcohol.  Similar  phenomena  are  met  with  in 
organised  nature,  where  two  liquids,  such  as  cell-sap  and  water,  are 
separated  by  a  membrane,  and  they  received  the  name  of  osmotic 
phenomena  (oxrpios — an  impulse).  M.  Traube  (1867)  showed 
that  artificial  semipermeable  septa  could  be  produced  from  such 
slimy  precipitates  as  gelatine  tannate,  cupric  and  zinc  ferro- 
cyanides,  and  W.  Pfeffer  (1877)  succeeded  in  depositing  these  in 
the  walls  of  a  porous  jar,  which  when  filled  with  a  solution  of 
salt,  or  sugar,  etc.,  closed  by  a  cork  through  which  passed  a 
mercury  manometer,  and  plunged  into  pure  water,  furnished  a 
means  of  measuring  the  osmotic  pressures.  By  the  use  of  such 
an  apparatus  ("  osmorneter  "),  or  modifications  of  it,  quantitative 
measurements  have  been  made  by  Pfeffer,  Adie,  Morse,  Lord 
Berkeley  and  others.  These  are  described  in  treatises  on  physical 
chemistry ;  we  shall  here  confine  our  attention  to  the  theory  of 
the  subject. 

Definition  of  Osmotic  Pressure. 

Let  a  given  liquid  solution  (e.g.,  a  solution  of  sugar  in  water) 
be  separated  from  the  pure  liquid  solvent  by  a  fixed  rigid 
diaphragm,  permeable  only  to  the  latter.  If  -,  ->'  are  the 
pressures  which  must  be  applied  to  solvent  and  solution,  respec- 
tively, to  maintain  equilibrium,  then  : 


280 


THERMODYNAMICS 


TT 


is  defined  as  the  osmotic  pressure  of  the  solution  at  the  (uniform) 
temperature  of  the  system,  and  with  the  solvent  under  the 
pressure  TT. 

If  the  septum  is  not  rigidly  fixed  to  the  cylinder  bounding  the 
liquids,  it  will  be  necessary,  in  order  to  maintain  equilibrium,  to 
apply  to  it  a  pressure  equal  to  TT'  —  IT,  or  P.  in  a  direction  from 
left  to  right.  For  the  pressure  TT  is  transmitted  unchanged 
through  the  septum  to  the  fluid  on  the  right,  and  if  an  additional 
pressure  TT'  —  TT  is  put  on  the  latter  by  means  of  the  septum, 
the  right  hand  piston  remains  in  equilibrium.  If  the  septum 
is  fixed,  this  force  is  exerted  on  it  by  reaction  from  the  fluid,  and 
becomes  evident  in  Nollet's  experiment. 

It  is  a  consequence  of  the  principle  of  Dissipatien  of  Energy 
that  P  is  positive  for  all  concen- 
trations   of  the   solution   and   all 
temperatures.     For  if  we  suppose 
the    end    pistons   fixed,   and    the 
•  septum   moved  towards  the  solu- 
tion, there  will  be  a  separation  of 
the  latter  i  nto  : 
(solvent)  +  (more  concentrated 

solution), 

and  the  work  (TT'  —  77)8  V  will  be 
spent  on  the  system,  where  8V  is  the  volume  through  which 
the  piston  advances.  Since  this  process  involves  the  reversal 
of  a  spontaneously  occurring  process,  viz.,  the  mixing  of  solvent 
with  solution  by  diffusion,  the  work  spent  is  positive,  hence  TT'  —  TT, 
or  P,  is  always  positive. 

We  shall  now  prove  that  P,  for  fixed  values  of  TT  and  the 
temperature,  is  definite  for  a  given  solution.  For  this  purpose 
we  have  first  of  all  to  show  that  the  dilution  or  concentration  of 
the  solution  can  be  effected  isothermally  and  reversibly.  If  the 
above  apparatus  is  constructed  of  some  good  conductor  of  heat, 
placed  in  a  large  constant-temperature  reservoir,  and  if  all  pro- 
cesses are  carried  out  very  slowly,  the  isothermal  condition  is 
satisfied.  Further,  suppose  the  end  pistons  fixed,  and  then  apply 
to  the  septum  an  additional  small  pressure  ^5P  towards  the 
solution.  There  will  be  a  slight  motion  of  the  septum,  through 
a  small  volume  8V,  and  work 
(P- 


ELEMENTARY  THEORY  OF  DILUTE    SOLUTIONS     281 

will  be  spent  on  the  system  in  separating  pure  solvent.  But  if 
the  pressure  on  the  septum  is  reduced  by  an  infinitesimal  amount 
8P,  there  will  be  a  slight  motion  in  the  opposite  direction,  and 
work 

(P  -  i8P)8V 
will  be  done  by  the  system. 

Since  SP  can  be  made  as  small  as  we  please,  the  operation 
becomes  in  the  limit  reversible. 

It  is  a  necessary  consequence  of  the  reversibility  of  osmotic 
processes  that  the  osmotic  pressure  is  independent  of  the  nature 
of  the  septum  used  to  measure  it.  For,  suppose  there  are  two 
semipermeable  septa  [a]  and  [/3],  and  let  the  osmotic  pressures  of 
a  solution  when  separated  from  pure  solvent  under  a  given 
pressure  by  these  septa  be  Pa  and  P^.  Then  if  we  separate  a 
volume  8V  of  solvent  through  [a],  the  work  Pa  8V  is  spent  on  the 
system,  and  if  the  solvent  is  readmitted  through  [3]  the  work 
P08V  is  done  by  the  system.  The  isothermal  cycle  being  now 
completed,  we  have  : 

-  P.8V  =  0 


Leakage  of  solute  through  an  imperfect  septum  would  constitute 
an  irreversible  phenomenon  (diffusion),  and  no  equality  of  Pa,  P^ 
need  then  result.  It  was  formerly  believed  that  the  osmotic 
pressure  of  a  solution  depended  on  the  nature  of  the  septum,  but 
this  was  merely  a  consequence  of  the  use  of  imperfect  septa  in 
the  experiments.  The  experimental  investigation  of  such  septa 
is  a  matter  of  great  practical  interest  and  importance,  but  is  of  no 
more  significance  in  the  theory  of  the  subject  than  is  leakage  of 
steam  in  the  cylinders  of  actual  engines  in  the  consideration  of 
the  expansion  of  gases. 

We  shall  now  pass  on  to  a  study  of  the  laws  of  osmotic  pressure, 
taking  up  in  the  first  instance  the  very  important  case  of  dilute 
solutions.  In  this  section  it  is  assumed  that  there  is  no  change 
of  total  volume  when  a  solution  is  diluted  by  further  addition  of 
pure  solvent,  and  that  solution  and  solvent  are  practically  incom- 
pressible. The  reader  will  then  easily  see  that  the  osmotic 
pressure  in  such  a  case  is  independent  of  the  pressure  supported 
by  the  pure  solvent  ;  the  complete  investigation  is  taken  up 
by  A.  W.  Porter,  Proc.  Roy.  Soc.,  A,  79,  519,  1907  ;  80,  457, 
1908. 


282 


THERMODYNAMICS 


128.     The  Laws  of  Osmotic  Pressure  for  Dilute  Solutions. 

We  assume,  on  the  basis  of  experimental  results,  that : 

(1)  The  volume  of  the  solution  obtained  by  mixing  a  volume 
TI  of  pure  solvent  with  a  volume  r2  of  a  dilute  solution  is  (i\  +  y2). 

This  is  usually  expressed  by  saying  that  the  volume  of  the 
solute  in  a  dilute  solution  is  independent  of  the  total  volume  of 
the  solution.  We  might  equally  well  say  that  the  volume  of  the 
solute  is  either  zero,  or  equal  to  the  total  volume  of  the  solution. 

(2)  If  a  dilute  solution  is  mixed  with  pure  solvent,  without 
performance  of  external  work,  there  is  no  evolution  or  absorption 
of  heat,  so  that  AU  =  0. 

Corollary. — The  work  done  in  the  isothermal  and  reversible 
dilution  of  a  dilute  solution  is  equal  to  the  heat  absorbed  from 
the  constant  temperature  reservoir. 

The  above  result  is  often  expressed  by  saying  that  the  intrinsic 
energy  of  the  solute  in  a  dilute  solution  is  independent  of  the 
volume. 

(3).  If  a  gas  is  in  isothermal  equilibrium  with  its  solution  in  a 
. ,      liquid,  the   concentration  in   the  solution  is 
proportional    to    the    pressure    of    the    gas 
(Henry's  law). 

In  the  original  investigation  of  van't  Hoff  (Zeitschr. 
physikal.  Chem.,  1,  1880;  Phil.  May.,  26,  1888),  the 
laws  of  dilute  solutions  are  arrived  at  separately, 
but  the  deduction  of  the  proportionality  between 
osmotic  pressure  and  concentration,  as  given  by  van't 
Iloff,  is  rather  an  analogy  than  a  stringent  proof, 
since  it  makes  use  of  hypothetical  considerations  as 
to  the  cause  of  osmotic  pressure.  The  following  proof, 
due  to  Lord  Bayleigh  (1897),  is  quite  strict,  and  has 
the  advantage  of  leading  directly  to  the  whole  theory. 
(Rayleigh,  Nature,  55,  253,  1897  ;  Donnan,  ibid. ;  cf. 
Larmor,  Phil.  Trans.  A,  190,  205,  1897.) 

We  shall  suppose  the  solute  to  be  a  mol 
of  an  ideal  gas,  occupying  a  volume  r  at  the 
pressure  p$,  and  the  solvent  a  volume  V  of 
liquid  just  sufficient  to  dissolve  all  the  gas 
under  the  pressure  j?o-  If  the  gas  is  brought  directly  into  contact 
with  the  liquid,  an  irreversible  process  of  solution  occurs,  but  if  it 
is  first  of  all  expanded  to  a  very  large  volume,  the  dissolution  may 
be  made  reversible,  except  for  the  first  trace  of  gas  entering  the 


V 

6 

/  1  '  •'•','  '  '  ' 

////y////' 

IP^ 

7 

FIG.  56. 


ELEMENTARY  THEORY  OF  DILUTE   SOLUTIONS    283 

liquid,  because  there  is  a  gas  pressure  corresponding  with  every 
concentration  of  the  solution,  however  small,  which  is  determined 
by  Henry's  law.  The  gas  can  then  be  slowly  compressed  so  that 
dissolution  proceeds  isothermally  and  reversibly  under  con- 
tinuously increasing  pressure,  until  the  last  trace  goes  into 
solution  under  the  pressure  pQ. 

Let  the  gas  and  liquid  be  contained  in  a  cylinder  of  unit 
cross-section,  fitted  with  pistons  and  fixed  diaphragms  as  shown 
in  Fig.  56.  a  is  an  impermeable  piston,  /3  a  fixed  diaphragm 
permeable  to  gas  but  not  to  liquid  (the  upper  surface  of  a  non- 
volatile liquid  satisfies  this  condition),  7  is  a  piston  permeable 
to  liquid,  but  not  to  dissolved  gas,  and  8  is  an  impermeable 
diaphragm  which  can  be  put  over  @  or  7  as  desired. 

Now  carry  out  the  following  isothermal  and  reversible  cycle  : 

(1)  Expand  the  gas  to  a  volume  x,  which  is  very  large  compared 
with  r,  keeping  8  over  yS. 

The  pressure  at  any  stage  (x)  of  the  expansion  is 


and  the  total  work  done  by  the  gas  on  expansion  is 


(2)  Remove  8,  and  let  the  rarefied  gas  begin  to  dissolve  in  the 
liquid,  pressing  down  a  reversibly.  The  pressure  on  a  is,  in  any 
given  position,  less  than  before,  because  some  gas  is  now  in 
solution.  It  follows  from  assumption  (1)  at  the  beginning  of 
this  section,  that  the  pistons  ft  and  7  may  be  kept  fixed,  and 
no  work  is  done  by  them.  The  value  of  the  gas  pressure  for 
the  position  a-  has  been  calculated  in  §  126,  (5),  to  be  : 


A,  being  the  solubility  of  the  gas.    p  is  therefore  a  function  of  .r. 
The  work  done  by  the  system  during  the  whole  second  operation  is  : 


r  ,1        r 

_  p(h.  =  _  Pov 

^    0  Jo 


+ 


__  _  o  ___. 

o 

The  total  work  spent  during  operations  (1)  and  (2)  is  therefore 
x  +  AV  ri  f     x  +  AY    , 


284  THERMODYNAMICS 

x  4-  AY 
By  hypothesis,  r  =  AV,  and  since  -  becomes  more  and 

more  nearly  equal  to  unity  as  .r  becomes  larger  and  larger,  it 
follows  that  the  (/as  has  been  dissolved  rcrersibli/  without  loss  or 
gain  of  work. 

(3)  To  complete  the  cycle,  we  must  get  the  gas  out  of  solution, 
and  restore  it  to  its  initial  state,  by  an  osmotic  process.  Raise 
a  and  7  simultaneously  through  the  spaces  r  and  V  respec- 
tively, so  that  the  solution  maintains  a  constant  composition 
throughout,  and  the  gas  and  osmotic  pressures  are  constant. 
The  work  done  is  }w  —  PY,  and  since  the  whole  work  in  the 
cycle  vanishes : 

Por  =  PY. 
But  2W  =  RT 

/.  PY  =  RT (1) 

where  T  is  the  temperature  at  which  the  isothermal  process  is 
executed. 

If  we  start  with  n  mols  of  gas  we  have 

p0v  =  nRT 

/.  PV  =  «RT          ....     (2) 
We  may  suppose  the  cycle  carried  out  at  various  temperatures, 
TI,  T2,  .  .  T.,,  and  so  obtain 

P!\T  =  RT1( 
P2Y  =  RT2, 

PrV  =  RT,, 
so  that  (1)  and  (2)  apply  quite  generally. 

It  therefore  appears  that  the  osmotic  pressure  is  : 

(1)  proportional  to  »/V,  i.e.,  to  the  concentration  f,  at  constant 

temperature ; 

(2)  proportional    to   the    absolute    temperature    at    constant 

concentration  ; 

(3)  the  same  for  equimolecular  concentrations  of  all  solutes, 

and  is  independent  of  the  nature  of  the  solvent ; 

(4)  equal  to  the  pressure  which  would  be  exerted  if  the  solute 

occupied  the  space  Y  in  the  condition  of  an  ideal  gas. 
We  may  therefore  sum  up  the  results  in  the  statement  that  the 
laws  of  osmotic  pressure  of  a  dilute  solution  are  formally  identical 
with  tJie  laws  of  qas  pressure  of  an  ideal  gas  (van't  HofTs  Gaseous 
Theory  of  Solution). 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS    285 

129.     Remarks  on  the  Theory  of  Solution. 

It  has  been  assumed  in  the  deduction  of  (1)  that  the  solute  is  an  ideal  gas, 
or  at  least  a  volatile  substance.  The  extension  of  the  result  to  solutions  of 
substances  like  sugar,  or  metallic  salts,  must  therefore  be  regarded  as 
depending  on  the  supposition  that  the  distinction  between  "  volatile  "  and 
"  non-volatile  "  substances  is  one  of  degree  rather  than  of  kind,  because  a 
finite  (possibly  exceedingly  small)  vapour  pressure  may  be  attributed  to 
every  substance  at  any  temperature  above  absolute  zero.  This  assumption 
is  justified  by  the  known  continuity  of  pressure  in  measurable  regions,  and 
by  the  kinetic  theory  of  gases. 

It  is  also  assumed  that  the  solute  does  not  change  its  molecular  weight  on 
passing  into  solution,  that  is,  does  not  polymerise  or  dissociate.  If  this 
were  the  case,  n  in  the  two  states  would  be  different,  and  if  1  mol  of  gas 
on  passing  into  solution  gave  rise  to  t  mols  of  solute,  we  should  have 

PV  =  iRT (3) 

where  i  is  known  as  van't  Hop's  factor. 

Lord  Kelvin  has  pointed  out  that  the  similarity  between  the  laws  of  gases 
and  of  dilute  solutions  carries  with  it  no  inference  as  to  physical  similarity 
between  the  states,  although  Boltzmann  has  developed  a  theory  of  osmotic 
phenomena  which  regards  the  pressure  as  due  to  a  bombardment  of  the 
semipermeable  wall  by  the  molecules  of  dissolved  solute,  whilst  it  is 
subjected  to  equal  and  opposite  forces  by  bombardment  from  the  solvent 
molecules  inside  and  outside. 

This  kinetic  explanation  of  osmotic  pressure  has  recently  received  experi- 
mental support  in  the  beautiful  researches  of  Perrin  on  the  Brownian  move- 
ment (Brcwnian  Movement  and  Molecular  Jieality,  J.  Perrin,  trans.  F.  Soddy, 
1910). 

The  kinetic  theory  of  gases  shows  that  the  pressure  p  exerted  by  a  gas  is 
given  by : 

pv  =  fnE  .     (1) 

where  H  is  the  number  of  molecules  in  a  volume  v,  and  E  is  the  mean 
kinetic  energy  of  a  molecule. 

If  we  now  assume  that  the  osmotic  pressure  of  a  solution  has  its  urigiu  in 
the  bombardment  of  the  semipermeable  diaphragm  by  the  molecules  of  the 
solute,  we  shall,  if  the  same  reasoning  is  applied  which  led  to  (1),  obtain  : 
Pr  =  fnE' 

where  E'  is  is  the  mean  kinetic  energy  of  a  molecule  of  solute.     If  «  is  the 
number  of  molecules  per  unit  volume : 

P  =  f»E' (2) 

Now  it  is  a  consequence  of  the  experiments  of  Pfeffer,  which  proved  that 
the  osmotic  pressure  was  equal  to  the  pressure  which  would  be  exerted  by 
the  same  number  of  molecules  of  an  ideal  gas  occupying  the  volume  of  the 
solution,  that : 

E  =  E' (3) 

so  that  the  ilissolveil  molecule  has  the  same  mean  kinelif  energy  as  if  it  existed  as 


286  THERMODYNAMICS 

a  yas  at  the  same  temperature.  This  kinetic  energy  being  proportional  to  p  at 
constant  volume,  is  proportional  to  the  absolute  temperature. 

Perrin  found  that,  if  an  emulsion  of  gamboge  were  allowed  to  settle,  the 
granules  did  not  all  fall  flat  to  the  bottom  of  the  vessel,  but  remained  per- 
manently forming  a  kind  of  atmospheric  haze  extending  to  a  short  distance 
into  the  liquid.  The  suspended  particles  were  seen  under  the  microscope  to 
be  in  Brownian  motion. 

Imagine  an  emulsion  formed  of  suspended  particles  all  exactly  alike,  and 
contained  in  a  vertical  cylinder  of  unit  cross-section.  The  state  of  a 
horizontal  slice  contained  between  the  levels  h  arid  h  -f-  dh  would  not  be 
changed  if  we  enclosed  it  between  two  semipermeable  pistons  which  allow 
the  molecules  of  water  to  pass,  but  stop  those  of  the  gamboge.  Each  piston 
experiences  an  osmotic  pressure,  by  reason  of  its  bombardment  by  gamboge 
particles,  and  if  there  are  /(  particles  per  unit  volume  at  height  /< 

l\  =  %nW 
and  Ph+<jlh  =  f(»  +  rfn)iE'. 

In  order  that  the  suspended  particles  shall  not  fall  through  the  liquid,  the 
effective  iveight  of  the  granules  must  be  equal  to  the  difference  of  osmotic 
pressures.  If  <p  is  the  volume  of  each  granule,  A  its  density,  and  8  that  of 
the  liquid,  we  have  therefore : 

-  |E'  dn  =  nrfA0(A  -  % 

or  -  fE^  =  ^ (A  -  S)ydh 

where  g  is  the  acceleration  of  gravity. 

By  integration,  we  obtain  the  following  relation  between  the  concentrations 
«0,  n  at  two  points  for  which  the  difference  of  level  is  h  : 

lWln'±=+(*-t)gh (3) 

so  that  the  concentration  of  the  granules  of  a  uniform  emulsion  decreases  in 
an  exponential  manner  as  a  function  of  the  height,  in  exactly  the  same  way 
as  the  barometric  pressure  as  a  function  of  the  altitude. 

Perrin,  by  a  series  of  researches,  measured  all  the  quantities  in  (3) 
directly,  except  E'.  He  then  found  that  the  exponential  law  did  really  hold, 
and  further  that  E'  had  the  same  order  of  magnitude  as  the  molecular  energy 
of  a  gas.  Since  the  gas  laws  hold  for  such  immensely  large  particles  as  the 
visible  granules  of  an  emulsion,  the  gaseous  theory  of  osmotic  pressure  is, 
apparently,  put  beyond  all  doubt  for  dilute  solutions. 

From  his  researches,  Perrin  calculated  the  number  N,  of  molecules  in  a 
mol  of  any  substance,  obtaining  the  value 

N  =  70-5  X  1022, 

which  agrees  moderately  well  with  the  estimations  of  other  recent  observers, 
who  used  entirely  different  methods  ;  the  mean  of  the  latter  is  60*9  x  1022. 

As  to  the  particular  way  in  which  solvent  manages  to  creep  through  the 
membrane,  to  the  exclusion  of  solute,  nothing  definite  can  be  said.  We 
might  assume  that  the  action  of  the  membrane  is  of  the  character  of  a  sieve, 
the  smaller  solvent  molecules  getting  through  its  pores  whilst  the  larger 


ELEMENTARY  THEORY  OF  DILUTE   SOLUTIONS    287 

solute  molecules  are  stopped.  In  the  first  instance  the  membrane  will  be 
bombarded  by  solvent  molecules  more  frequently  on  the  side  of  pure  solvent 
than  on  the  solution  side,  because  some  of  the  impacts  from  the  solution 
will  be  due  to  molecules  of  solute.  Solvent  will  therefore  pass  into  the 
solution  until  its  concentration  is  equal  to  that  outside,  and  every  part  of 
the  membrane  is,  on  the  average,  Istruck  equally  frequently  by  solvent 
molecules  on  both  sides.  It  might  be  objected  that,  if  the  solute  molecules 
screen  the  membrane  from  impacts  on  the  inside,  they  will  equally  well 
retard  the  entrance  of  solvent  molecules  from  outside,  but  if  we  regard  each 
solute  molecule  as  having  about  it  a  kind  of  "  sphere  of  influence  "  or  loose 
colony  of  molecules  of  solvent,  an  entering  molecule  of  solvent  might  dis- 
lodge a  similar  molecule  from  this  assemblage,  and  take  its  place.  The 
inability  of  solute  molecules  to  pass  through  the  membrane  might  also  have 
its  origin  in  the  comparatively  large  si/e  of  the  collection  of  solvent  mole- 
cules clustered  about  it.  The  passage  of  solvent  through  the  membrane  may 
not  be  a  simple  mechanical  transit  through  cavities  or  tunnels  in  the  latter, 
but  may  be  more  of  the  type  of  action  contemplated  by  Graham  in  his 
theory  of  atmolysis.  Carbon  dioxide  was  found  to  pass  more  readily  than 
air  through  a  rubber  membrane,  and  Graham  thought  this  might  be  explained 
if  we  assume  that  the  gas  dissolves  in  the  portion  of  membrane  presented  to 
it,  diffuses  through  the  body  of  the  membrane,  and,  after  soaking  through, 
re-evaporates  on  the  other  side.  However  the  actual  passage  of  solvent 
through  the  membrane  and  the  physical  cause  of  osmotic  pressure  are  to  be 
explained,  the  basis  of  the  similarity  between  the  laws  of  osmotic  pressure 
and  the  laws  of  gases  is  quite  clear.  The  solute,  when  it  goes  into  solution , 
is  not  distributed  continuously  throughout  the  solvent,  but  is  divided  up 
into  discrete  portions,  which  may  be  all  alike,  or,  when  there  is  an  equilibrium 
between  portions  of  greater  and  less  molecular  complexity  ("association," 
or  "  dissociation  "—whether  simple  or  electrolytic)  may  be  of  different  kinds. 
Each  discrete  portion  of  solute,  which,  with  the  above  extension,  may  be 
called  a  "  molecule  "  of  solute,  influences  in  some  way,  the  exact  character 
of  which  is  immaterial  for  the  present  theory,  the  neighbouring  molecules 
of  solvent.  From  what  we  know  (from  capillary  phenomena  and  otherwise) 
of  the  nature  of  molecular  forces,  this  range  of  influence  must  be  very 
minute,  so  that  if  the  solution  is  dilute,  these  regions  of  influence  do  not 
intersect,  but  are  separated  by  unaltered  solvent.  Each  is  therefore,  for  a 
very  large  part  of  any  time,  quite  independent  of  the  others,  and  has  a 
mean  free  path.  We  have  thus  a  state  of  affairs  exactly  comparable, 
although  by  no  means  physically  identical,  with  the  state  of  a  gas.  Any 
further  addition  of  solvent  then  serves  merely  to  separate  the  regions  of 
influence  to  greater  distances,  and  all  considerations  as  to  the  mode  of 
influence  of  the  solute  on  the  solvent,  such  as  the  extent  or  manner  of 
hydration  of  the  former,  are  not  germane  to  the  problem.  This  point  is  all 
the  more  to  be  emphasised  on  account  of  the  feeling  which  unfortunately 
seems  to  be  widely  spread,  that  the  older  chemical  hydrate  theory  of  solution 
is  opposed  to  the  new  "  gaseous  theory  "  of  van't  Hoff.  This  has  apparently 
arisen  from  the  one-sided  views  adopted  by  the  champions  of  each  theory, 
and  is  certainly  not  a  necessary  consequence  of  the  fundamental  basis  of 


288 


THERMODYNAMICS 


either.  The  real  fundamental  proposition  of  the  thermodynamic  theory  of 
solution  is  that  the  osmotic  pressure  of  a  solution,  and  every  other  pro- 
perty conditioned  solely  by  it,  depend  simply  on  the  number  of  solute  mole- 
cules scattered  through  a  given  volume  of  solution,  and  not  at  all  on  the 
chemical  nature  of  either  solute  or  solvent,  or  on  the  relation  between  the 
latter,  provided  only  that  the  solution  is  dilute.  These  properties  are  there- 
fore colligative  or  molar.  The  chemical  properties  of  the  solution,  on  the 
contrary,  depend  not  only  on  the  number,  but  also  on  the  nature,  of  the 
dispersed  particles,  and  so  are  to  a  large  extent  conditioned  by  the  exact 
mode  of  connexion  between  the  solute  and  solvent.  It  is  greatly  to  be 
desired  that  writers  on  the  theory  of  solution  should  distinguish  clearly 
exactly  which  aspect  of  the  matter  belongs  properly  to  their  own  investiga- 
tions, and  refrain  from  attacking  on  the  basis  of  irrelevant  experiments  a 
theory  which  is  quite  immune  from  the  criticism  which  may  reasonably  be 
levelled  against  any  particular  hypothetical  view  of  the  nature  of  solutions. 
The  thermodynamic  aspect  of  osmotic  pressure  is  to  be  sought  in  the 
expenditure  of  work  required  to  separate  solvent  from  solute.  The  separa- 
tion may  be  carried  out  in  other  ways  than  by  osmotic  processes ;  thus,  if 
we  have  a  solution  of  ether  in  benzene,  we  can  separate  the  ether  through  a 
membrane  permeable  to  it,  or  we  may  separate  it  by  fractional  distillation, 
or  by  freezing  out  benzene,  or  lastly  by  extracting  the  mixture  with  water. 
These  different  processes  will  involve  the  expenditure  of  work  in  different 
ways,  but,  provided  the  initial  and  final  states  are  the  same  in  each  case, 
and  all  the  processes  are  carried  out  isothermally  and  reversibly,  the 
quantities  of  work  are  equal.  This  gives  a  number  of  relations  between  the 
different  properties,  such  as  vapour  pressure  and  freezing-point,  to  which  we 
now  turn  our  attention. 


130.     Vapour  Pressures  of  Dilute  Solutions  :   Thermodynamic 
Theory. 

Let  a  dilute  solution  of  a  (volatile  or  involatile)  solute  in  a 
volatile  solvent  be  contained  within  the 
semipermeable  walls  a,  b,  separating  it 
from  pure  solvent,  and  the  vapour  of 
the  latter  emitted  from  the  solution, 
respectively.  Above  is  a  cylinder  C, 
in  which  the  pressure  of  the  vapour 
can  be  brought  to  any  desired  value. 
The  whole  apparatus  is  supposed  to  be 
free  from  gravitational  action,  and  the 
following  isothermal  reversible  cycle  is 

.T  llr.     Ot  .  ,        T 

executed : 

(1)  Take  a  small  volume  rA  of  vapour  from  the  solution  under 
the  vapour  pressure  p' ' , 


ELEMENTARY  THEORY  OF  DILUTE   SOLUTIONS    289 


(2)  Compress   the  vapour   in   C    to   pressure  p,   the  vapour 
pressure  of  the  pure  solvent,  and  let  its  volume  now  be  rB. 

(3)  Condense  the  vapour  on  the  solvent  under  the  pressure  p. 

(4)  Allow  the  added  solvent,  of  volume  V,  to  pass  reversibly 
through  a  into  A,  against  the  osmotic  pressure  P  of  the  solution 
(with  the  pure  solvent  under  the  pressure  of  its  own  vapour). 

If  we  neglect  the  very  small  amounts  of  work  done  during  the 
changes  of  volume  of  the  liquids  when  vapour  is  removed  from 
or  condensed  upon  them,  the  net  work  in  the  cycle  is  : 


A  -  KB  + 


+  PV  =  0. 


But,  from  the  rule  of  integration  by  parts  : 


fpdv 

J  v. 


=  pi-B  —  p'r±  —     r 

^   P 

.  PV  =  I  vdp       . 

J  •,,! 


(1) 


127) 


Corollary  1. — Since  P  is  essentially  positive 

p'  <  p     •         •         •         •         •     (2) 

so  that  the  vapour  pressure  is  less  over  the  solution  than  over 
pure  solvent,  at  the  same  temperature. 

It  follows  directly  from  this  that  the  vapour  pressure  of  a 
solution  is  lower,  at  a  given  tempera- 
ture, than  that  of  pure  solvent.  For, 
if  temperature  and  pressure  are  co- 
ordinates, a  horizontal  line  through 
p  =  1  atm.  will  cut  the  vapour  pressure 
curves  of  solution  and  solvent  at  points, 
the  abscissae  of  which  represent  tempera- 
tures at  which  both  vapour  pressures 
are  equal  to  atmospheric  pressure,  i.e., 
the  boiling-points. 

That  this  value  of  T  is  greater  for  the 
solution  than  for  the  solvent,  follows  from  the  fact  that  the  curve 
of  the  latter  is  intersected  first,  for  the  vapour-pressure  curve  of 
the  solution  (s's)  must  lie  beneath  that  of  the  pure  solvent  (ss) 
in  the  vicinity  of  the  boiling-point.  The  corollary  (1)  to  equa- 
tion (7)  below  extends  this  conclusion  over  the  whole  length  of 


FIG.  58. 


290  THERMODYNAMICS 

the  curve.     T,  T'  represent  temperatures  at  which  pure  solvent 
and  solution  have  the  same  vapour  pressure  p'  ,  and  : 
T'  >  T. 

We  now  assume  that  : 

(i.)  The  vapour  of  the  solvent  obeys  the  gas  laws. 

(ii.)  The  osmotic  pressure  obeys  the  laws  of  dilute  solutions. 

Then  prB  —  »0BT         .        ..        .         .     (3) 

where  n0  =  number  of  mols  of  vapour  in  the  volume  rB  ; 
and  PV  =  ?iET        ....     (4) 

where  n  =  number   of   mols   of   solute   in   the   volume   V   of 
solution. 

From  (1),  (3),  and  (4)  we  obtain  the  relation  : 


But  In    ,  -  lnl  +  =  -  (6) 

P  \  P      I  P 

approximately,  if  p  —  p'  is  small  compared  with  p. 

Thence 


or  '- £-  = — V-          •         •         •     (7) 

p  -no  +  » 


Corollary  1.— The  relative  lowering  of  vapour  pressure 

i' 
is  independent  of  temperature  (Law  of  von  Babo). 

Thus  the  vapour-pressure  curves  of  the  solution  and  solvent 
are  similar  and  similarly  situated,  i.e.,  if  we  know  the  form  of 
the  vapour -pressure  curve  of  the  pure  solvent,  those  of  all  the 
solutions  are  also  known. 

Corollary  2. — The  relative  lowering  of  vapour  pressure  is  pro- 
portional to  the  concentration  c  —  n/  (//o  +  »)  at  all  temperatures, 
so  that  if  equimolecular  amounts  of  different  substances  are  dis- 
solved in  equal  weights  of  solvent,  the  solutions  all  have  the 
same  vapour  pressure,  independently  of  the  natureof  the  solute. 

Corollary  3. — If  solutions  are  prepared  with  different  solvents 
and  solutes  so  that  in  all  cases 

number  of  mols  of  solute 

total  number  of  mols  in  solution 

has  the  same  value,  the  relative  lowering  of  vapour  pressure  will 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS    291 

be  the  same  in  all  cases,  and  numerically  equal  to  this 
fraction. 

These  results  were  all  experimentally  verified  by  Raoult,  who 
found  the  value  O0104  for  the  mean  molecular  lowering  with  12 
solvents  and  a  variety  of  organic  solutes,  in  a  solution  with 

u    4. ,}  ~  -mo'  w^ereas  (7)  giyes  O'Ol.     The  following  figures  for 

cane  sugar  at  0°  in  aqueous  solution,  by  Dieterici  (1897),  and 
Srnits  (1904),  show  how  good  is  the  agreement  when  we  take 
account  of  the  fact  that  (_/;  —  p')  was  always  less  than  0*05  mm. 


Mols  per 
l.OOOgr.  H-A 

n  mols  per 
100inolsH2O. 

(P  ~  P'}  4-  n. 

(»-*'  •  n\ 

\     p 

molec.  lowering. 

0-0509 

0-0916 

0-0467 

0-0102 

0-1506 

0-2711 

0-0467 

0-0102 

0-1723 

0-3101 

0-0477 

0-0104 

0-2653 

0-4775 

0-0467 

0-0102 

0-4541 

0-8174 

0-0483 

0-0105 

0-4993 

0-8987 

0-0483 

0-0105 

1-0098 

1-8134 

0-0500 

0-0109 

1-0122 

1-8220 

0-0500 

0-0109 

It  will  be  noticed  that  we  make  no  assumption  as  to  the  mole- 
cular weight  of  the  solvent  in  the  liquid  state.  Equation  (3) 
refers  to  the  vapour  only.  It  is  to  be  expected,  therefore,  that 
when  the  solvent  does  not  yield  a  vapour  having  the  normal 
density,  the  value  of  the  molecular  lowering  will  be  abnormal. 
Eaoult  found  that  when  acetic  acid  was  used  as  solvent  the 
observed  molecular  lowering  was  0'0163.  Acetic  acid,  however, 
is  known  to  be  polymerised  in  the  state  of  vapour ;  at  the  boiling- 
point  the  molecular  weight  as  determined  by  the  vapour  density 
is  1'64  times  the  normal  (C2H402  =  60).  The  number  of  mols 
per  unit  volume  will  be  reduced  in  the  same  ratio,  and  hence  we 
must  write  (3) : 


u  2 


292 

and  (7) : 


THERMODYNAMICS 


.          X    :r  -A  =  0-0164 

«„  +  it       1'64 


very  approximately,  since  n   is  very  small   compared    with  n0. 
This  agrees  almost  exactly  with  the  observed  value. 

In  the  case  of  aqueous  salt  solutions,  the  observed  molecular 
lowering  was  invariably  greater  than  the  calculated.  In  this 
case  we  put  instead  of  (4) : 

p\  =  inET 


and  hence 


7        P  •    H 

In  * -,  =  i  - 
P  »o 


or 


.P 


approximately         .     (8) 


The  following  numbers  were  obtained  by  Smits  (1902) : 
Sodium  chloride  in  water  at  0°  C. 


Mols  per 
l,000gr.  H20. 

Mols  per 
100molsH2O(7i). 

1 

np 

i 

0-0591 

0-1064 

0-01743 

1-79 

0-0643 

0-1157 

0-01741 

1-76 

0-1077 

0-1939 

0-01727 

1-72 

0-1426 

0-2567 

0-01727 

1-72 

0-4527 

0-8149 

0-0170 

1-70 

0-4976 

0-8957 

0-0170 

T70 

1-0808 

1-9454 

0-01727 

1-723 

1-2521 

2-2538 

0-01740 

1-730 

Sulphuric  acid  in  water  at  0°  C. 


Mols  per 
l,000gr.  H2O. 

Mols  per 
100molsH.20(n). 

P-P' 

i 

np 

0-0951 

0-1712 

0-0204 

2-03 

0-1208 

0-2174 

0-0188 

1-87 

0-4215 

0-7587 

0-0193 

1-93 

0-9762 

1-7572 

0-0207 

2-06 

ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS     293 

The  relation  between  osmotic  pressure  and  vapour  pressure 
was  deduced  by  Gouy  and  Chaperon  (1888),  and  independently 
by  Arrhenius  (1889). 

Equation  (7)  is  true  for  volatile  as  well  as  involatile  solutes,  pro- 
vided n  denotes  the  number  of  niols  of  solute  in  the  liquid  phase, 
and  p'  is  the  partial  pressure  of  the  vapour  of  the  solvent,  the  latter 
being  independent  of  the  presence  of  other  gases  in  the  vapour 
space.  The  sole  remaining  problem  is  therefore  the  determina- 
tion of  the  partial  pressure  of  the  solute,  or,  what  will  lead  to 
this,  the  total  pressure  in  the  vapour  space.  The  partial  pressure 
of  the  solvent  is,  from  Raoult's  law  : 


and  if  TT  is  the  partial  pressure  of  the  solute,  the  total  pressure  is  : 

n  =  P  i^fT7,  +  T  •        •  (9) 

If  the  solute  is  a  gas,  -n  is  known,  for  a  fixed  total  volume, 
from  the  solubility  law  of  Dalton,  hence  FI  is  determined. 

Put  ii/(ii0  -f-  M)  =  e,  TT/U  =  c'  .  .  (10) 
for  the  concentrations  of  solute  in  the  liquid  and  vapour  phases, 
then: 


which  reduces  to  Raoult's  equation  as  a  special  case  for  a  non- 
volatile solute  (TT  =  0,  c'  =  0).  Equation  (11)  is  due  to  Nernst 
(1891). 

Corollary  1.  —  The  vapour  pressure  of  the  solution  is  greater 
than  that  of  the  pure  solvent  when  the  concentration  of  the 
solute  is  greater  in  the  vapour  than  in  the  liquid  phase. 

Corollary  2.  —  The  vapour  pressure  of  the  solution  is  equal  to 
that  of  the  pure  solvent  when  e  =  c'.  Since,  by  Henry's  law, 
c/c'  depends  only  on  temperature,  and  since  distillation  of  liquid 
cannot  alter  its  composition  in  this  case,  the  solution  will  distil 
unchanged  at  a  constant  temperature  exactly  like  a  pure  substance. 
This  holds  only  within  the  limits  of  applicability  of  Henry's  law. 

If  several  different  solutes  are  present  together  : 


If  we  assume,  with  Nernst  (1893),  that  the  osmotic  pressure  of 
a  solute  (being  independent  of  the  nature  of  the  solvent  provided 


'294  THERMODYNAMICS 

no  molecular  change  results)  is  the  same  in  a  mixture  of  solvents 
as  in  a  pure  solvent,  then,  since  for  NX  mols  of  solvent  [1]  and 
N2  mols  of  solvent  [2]  per  mol  of  solute,  we  have  by  Raoult's  law  : 


In&r  =  w 
;>•>         JNo 


therefore  1  =  NI  In 1>lt  +  N2/n *~ ,         '.         .     (13) 

Pi  PS 

This  was  verified  by  Roloff  (1893).  Some  of  the  terms  on  the 
right  of  (13)  may  be  negative.  Thus,  if  potassium  chloride  is 
added  to  a  mixture  of  water  and  acetic  acid,  the  partial  pressure 

of  the  acetic  acid  is  raised  :  p%  >  ^2,  •'•  In—,  <  0,  which  implies 

that  the  solubility,  A,  of  acetic  acid  in  water  is  reduced  by  the 
addition  of  the  salt  (Nernst :  Thcoretische  Chemie). 

131.     Boiling-Points. 

Let  AA',  BB'  represent  portions  of  the  vapour-pressure  curves 
of  pure  solvent  and  solution,  respec- 
tively. Draw  PQ,  QR  parallel  to  the 
axes,  then 

PQ  =  RQ  tonPRQ 
.'.  p  —p'  =  ST  tawPRQ    .     (1) 
p  —p'    is    the     lowering     of     vapour 
pressure,  ST   the  elevation  of  boiling- 
point. 

— I       If  RQ  is  small,  we  have  by  the  mean 
value  theorem  (H.M.  §  69) : 

i 

=  TTp  for  the  pure  solvent. 
/.  p-p'  =5T^     .         .         .         .   (la) 

But  ^  =  - *£- =   p\     .         .         .    (i/,) 

with  the  assumptions  of  §  88. 

•••^£'  =  RW8T-  <2> 

If  p  =  1  atm.,  5T  is  the  elevation  of  boiling-point,  T0  is  the 
boiling-point  of  the  pure  solvent. 


ELEMENTARY   THEORY  OF   DILUTE   SOLUTIONS    295 

Corollary.  —  The  molecular  elevation  of  boiling-point  is  indepen- 
dent of  the  nature  of  the  solute. 

Ae  refers  to  a  mol  of  solvent,  /.  R  =  T985  g.  cal.,  or  2  g.  cal., 
approximately, 


But  *  -  —  =  —  ,  very  nearly, 

1>  n0 

.-.  for  1  mol  of  solute  in  100  niols  of  solvent, 

»T  =  °^W      .       .       .      .    (4) 

« 

in  which  the  molecular  elevation  of  boiling-point,  ST,  is  given 
entirely  in  terms  of  quantities  which  depend  only  on  the 
properties  of  the  pure  solvent. 

Equation  (4),  deduced  by  van't  Hoff  (1886),  was  verified 
experimentally  by  Beckmann  (1889). 

If  p  —  p'  is  too  large  to  justify  the  assumption  of  (la),  we  can 
integrate  (lb)  on  the  assumption  that  A,  is  constant  : 

Inp  =  —  ~  +  const. 

where  p  is  the  vapour  pressure  of  the  pure  solvent  at  the  boiling- 
point  T  of  the  solution.  At  the  boiling-point  of  the  pure  solvent  : 

Inp'  =  —  ^  +  const., 
since  p'  is  atmospheric  pressure, 


But 


From  the  results  of  §  130,  we  have  : 

~p 

PV  =  ;*oRT       ^  =  »»oBT/w  ^. 

JP,P  P 

If  Mo,  p  are  the  molecular  weight,  and  density,  of  the  solvent  : 

y       _        »OM0 

p 


296  THERMODYNAMICS 

To  express  P  in  atmospheres,  we  take  the  litre  as  unit  volume, 
aiid  put  R  =  0-08207  1.  atm. 

_  0-08207  X 
Mo 
From  (6)  and  (5)  we  obtain  : 


But  A,/M<,  =  L,,,  and  T  -  T0  =  5T, 


PT 
8T  =  pL;    •        •        •        '        •     <8) 

To  express  P  in  atmospheres,  we  put 

L, 


"  24-191 
and  use  (6a) : 

1000/>L,  5T    . 

P  =  ^19T  Tatm'      K|   '       "(8a) 


132.    Thermodynamic  Theory  of  Freezing-Points  of  Solutions. 

The  freezing-point  of  a  solution  is  the  temperature  at  which 
the  solution  is  in  equilibrium  with  ice,  the  latter  term  being  used 
in  its  general  significance  of  frozen,  or  solid,  solvent. 

The  vapour  pressure  of  the  solution  at  the  freezing-point  is 
equal  to  that  of  pure  ice  at  the  same  temperature  (Guldberg,  1870). 
For  if  we  take  the  system  :  ice,  solution,  vapour,  at  the  freezing- 
point,  and  suppose  that  p1 ',  p"  are  the  vapour  pressures  of  solution 
and  ice,  r',  v"  the  specific  volumes  of  the  vapour  under  these 
pressures,  and  V,  Y"  the  specific  volumes  of  solution  and  ice,  we 
may  execute  the  following  isothermal  cycle  : 

(1)  Evaporate  unit  mass  of  solvent  from  a  large  quantity  of 
solution. 

(2)  Compress  or  expand  the  vapour  until  it  is  in  equilibrium 
with  the  ice. 

(3)  Condense  the  vapour  on  the  ice. 

(4)  Allow   unit   mass   of    ice   to   melt    in    contact   with   the 
solution. 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS     297 


The  net  work  in  the  cycle  is  zero,  hence  : 

p'(v'  -  V)  +     pdv  +  p" (V"  -  v")  +  p"(T  -  V")  =  0 

Vdp  +  (p"  -  j/)V"  =  0 


.'.p"  =  p'  .       (1) 

Thus  the  vapour  -pressure  curves  of  solution  and  of  ice  must 
intersect  at  the  freezing-point.  But  the  vapour-pressure  curve  of 
the  solution  lies  below  that  of  pure 
solvent  throughout  its  length,  and 
since  the  latter  cuts  the  vapour- 
pressure  curve  of  ice  at  the  freez- 
ing-point of  the  pure  solvent,  the 
former  must  cut  the  vapour-pressure 
curve  of  ice  at  a  lower  temperature, 
or  in  other  words  the  freezing- 
point  of  the  solvent  is  depressed  by 
addition  of  solute.  It  is,  of  course, 
assumed  that  the  solid  which 
separates  is  pure  ice,  for  otherwise 
the  pressure  curve  of  the  solution 
would  not  cut  the  pressure  curve 
of  ice  at  the  freezing-point,  but  some  other  curve,  belonging  to  the 
separating  solid. 

Let  OA,  AS  represent  the  vapour-pressure  curves  of  the  ice  and 
liquid  solvent  respectively,  BS'  that  of  the  dilute  solution.     AC  is 
the   vapour-pressure   curve   of  supercooled  liquid.       T0 
freezing-point  of  pure  solvent,  T  that  of  the  solution. 
CS,  OA  we  have  : 

dp,  A,,  A,, 

cW  =  T(r,  -  r,)  = 

dp,  A  .  A, 


To 


FIG.  60. 


is   the 
Along 


±\10     —     V,)  LVg 

If  the  vapour  obeys  the  gas  laws  : 
dlnpe         \e 

dT~  =  RT* 
dlnp,  _    \s_ 

and  if  \e,  A,  do  not  vary  appreciably  with  temperature  over  a 


298  THERMODYNAMICS 

small   range,  which  presupposes  (T0  —  T)  to  be  small,  or   the 
solution  very  dilute,  we  have  : 

lnpe  =  —  —<  +  c', 

and  lnpx  =  -  ^  +  c"  .         .         .         .     (2) 

At  the  temperature  T0,  we  have 

Pe=P*=Po         •         -        •        •     (3) 
where  pQ  is  the  common  pressure  of  ice  and  solvent  at  A, 


and                                  /»7>o  =  —  ^?jr  +  «" 

I '  -a  -  _  ^  P-  — 1 

PQ             R  LT  ToJ 

and  In  —  =  —  : 


so  that  ^^^.         .         .(4) 

But  pe  =  p  =  vapour  pressure  of  supercooled  liquid  at  C, 

2>s  =p'  =    ,,         „         „  solution  at  its  freezing-point. 
Also  A,  —  A,,  =  Ay,  the  molecular  heat  of  fusion  (§  117), 


Equation  (5),  in  a  slightly  more  general  form,  was  deduced  by 
Guldberg  (1870).  The  latter  allowed  for  the  change  of  A,  with 
temperature,  which  has  been  neglected  above  on  account  of  its 
very  small  magnitude. 

If  A  is  in  calories,  K  =  T985. 

To  get  the  relation  with  osmotic  pressure,  we  have  : 

P  =  .4.  BT//I  >- 
Mo  p 


.    P_  f 

-  MO  LT~TJ=  To"8 

where  L  f  =  A7/M0  =  latent  heat  of  fusion  per  unit  mass, 

ST  =  TO  —  T  =  depression  of  freezing-point. 
To  express  P  in  atmospheres,   we  multiply  Ly  by  24*191  to 


ELEMENTARY   THEORY  OF   DILUTE    SOLUTIONS     299 

reduce  to  litre-atmospheres,  and  take  1000/3,  so  as  to  refer  to  the 
litre  as  unit  volume  : 

100QpL,ST 
"   24-191    To 
If  there  are  £  mols  of  solute  per  litre, 

.....     (8) 


-  - 

'    24-191  '  To' 

and  since  R  =  0'08207 


But  if  R  in  (8)  is  taken  as  1"985  g.  cal.,  L/must  be  in  g.  cal., 
and  we  still  obtain  (9). 

In  the  case  of  water  :  L,  =  80,  p  =  1,  T0  =  273 

.-.  8T  =  l-86£       ....     (10) 

which  gives  the  depression  for  £  mols  of  solute  per  litre  in  terms 
of  quantities  depending  on  the  pure  solvent  only. 

Raoult  expressed  his  results  in  terms  of  the  molecular  depres- 
sion (J3,  for  a  mol.  of  solute  in  100  grams  of  solvent.  The  volume 
of  the  solvent  is  100/p,  and  this  may  be  taken  as  the  volume  of 
the  dilute  solution.  The  corresponding  osmotic  pressure  P',  on 
the  assumption  that  the  law  of  proportionality  holds  good  at 
this  concentration  (which  is  only  a  fictitious  extrapolation)  is 
given  by  : 

P:F  =  10°:1000 

P 

.'.  P'  =  10pP      ....     (11) 
.-.  P'V  =  flOpRTo  from  (8). 

Thence  8T  =  o-oonTOxl(v 

;jV.;.;;:^:    =^5  .....  (12) 

very  approximately,  where  5  denotes  the  number  of  mols  of  solute 
per  100  gr.  solvent. 

If  5  =  1,  8T  =  <£,  the  molecular  depression, 

.'.*=5™y     ....     (18) 

This  equation  was  deduced  by  van't  Hoff  in  1885,  and  provides 
a  simple  method  of  determining  the  latent  heat  of  fusion  of  a 


300 


THERMODYNAMICS 


substance.  All  that  is  necessary  is  to  find  the  freezing- 
point  (T0)  of  the  pure  substance,  and  the  molecular  depression 
by  means  of  a  substance  of  known  molecular  weight  used  as 
solute. 

Eykman  has  verified  van't  Hoff's  equation  with  a  large  number 
of  substances  used  as  solvents.  The  values  of  Lf  calculated 
from  (13),  and  those  observed  directly,  are  given  below  in  a  few 
cases : 


- 

<*> 

0-02  T02/L/ 
obs. 

TO  -  273. 

L/  obs. 

L/  calc. 

Water 

18-5 

18-54 

0 

80 

!    78-75 

Formic  acid 

28 

27-5 

8 

57-38 

56-4 

Acetic  acid 

39 

38-5 

17 

43-66 

43-1 

Stearic  acid         .        45 

47-7 

64 

47-6 

50-5 

Benzene     .         ;" 

51-2 

51-9 

5-5 

29-9 

30-3 

2>toluidine 

53 

51 

42-1 

39 

37-5 

Phenol 

727 

78-6 

40 

24-93 

27*0 

Phenylacetic  acid 

90 

97 

79 

25-4 

27-5 

It  was  found  that  the  law  of  proportionality  (12)  holds  good  .only 
if  the  solution  is  dilute,  and  in  the  determination  of  </>  one 
usually  adopts  Eykman's  method  by  finding  the  molecular 
depressions  for  a  few  different  concentrations  in  dilute  solutions, 
plotting  these  against  the  concentrations,  and  extrapolating  to 
zero  concentration,  or  infinite  dilution  (H.M.,  §  67). 

Raoult  observed  that  many  substances  dissolved  in  benzene, 
nitrobenzene,  and  ethylene  dibromide,  gave  depressions  only  half 
the  normal,  and  this  he  explained  as  due  to  a  polymerisation  of 
the  solute  to  double  molecules  : 

2A  =  A2. 

In  confirmation  it  was  observed  that  such  substances  (e.g.,  acetic 
acid)  gave  abnormally  high  vapour  densities. 

But  when  solutions  of  salts  in  water  were  found  to  give  depres- 
sions considerably  in  excess  of  the  normal,  usually  approaching 
double  that  amount  at  high  dilution,  the  interpretation  was  by  no 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS     801 

means  clear.  We  might  suppose  that  all  the  so-called  normal 
depressions  produced  by  organic  solutes  are  really  due  to  double 
molecules,  and  that  the  salts  are  normal,  but  the  whole  body  of 
external  evidence  is  unanimously  against  this  hypothesis.  The 
only  other  explanation  possible,  if  we  suppose  the  law  of  propor- 
tionality to  be  valid  in  all  cases,  is  to  suppose  that  the  salts  are 
broken  up  in  solution,  and  Arrhenius  (1887)  showed  that  this  is 
also  in  agreement  with  the  electrical  properties  of  the  solutions. 
He  supposed  that  the  salt  molecules  break  down,  to  a  greater  or 
less  extent,  into  sub-molecules,  or  ions,  which  carry  electrical 
charges : 

KC1  =  K-  +  C1-, 

and  the  increased  number  of  molecules  of  solute  so  produced 
accounted  for  the  abnormally  large  depressions.  In  this  case,  if 
we  put : 

P=i£RT0         .         .         .         .     (8a) 

instead  of  (8),  where  i  is  the  ratio  of  the  observed  osmotic  pres- 
sure to  that  calculated  from  van't  Hoff's  theory,  we  shall  obtain 
instead  of  (12) : 

6T  =  ow 

"/ 

and  i  is  also  the  ratio  of  the  observed  to  the  calculated  depression 
of  freezing-point. 

A  similar  method  was  used  in  connexion  with  the  lowering  of 
vapour  pressure  in  §  130.  It  is  evident  that,  since  the  factor  / 
was  introduced  in  the  same  connexion  in  both  investigations,  the 
values  of  i  obtained  by  both  methods,  viz.,  by  measurements  of 
vapour  pressure  and  of  freezing-point,  are  necessarily  the  same, 
and  their  agreement  is  therefore  independent  of  any  theory  which 
we  may  adopt  to  explain  the  anomalous  behaviour  of  aqueous  salt 
solutions. 

The  test  of  the  validity  of  the  theory  of  Arrhenius  is  not 
therefore  to  be  found  in  the  agreement  between  the  values  of  » 
obtained  from  measurements  of  any  properties  of  solutions  which 
are  conditioned  by  the  osmotic  pressure ;  it  is  in  quite  another 
field — that  of  electrochemistry — that  a  comparison  of  known 
relations  with  the  deductions  from  the  theory  may  be 
instituted. 


302  THERMODYNAMICS 

The  following  numbers  are  given  by  Dieterici  (1891) : 


Solute. 

i  (vapour  pressure). 

i  (freezing-point). 

NaCl 

•92 

1-90 

KC1 

•78 

1-82 

KBr 

•74 

1-90 

KI 

•82 

1-90 

LiCl 

•92 

1-99 

NaN03 

•65 

1-82 

There  is  fair  agreement  between  the  two  sets,  considering  the 
difficulty  of  the  vapour-pressure  measurements.  . 

133.     Heat  of  Solution  ;   Effect  of  Temperature  on  Solubility. 

A  liquid  solution  may  be  separated  into  its  constituents  by 
crystallising  out  either  pure  solvent  or  pure  solute,  the  latter  pro- 
cess occurring  only  with  saturated  solutions.     (At  one  special 
temperature,  called  the  eryohydnc  tempera- 
ture, both  solvent  and  solute  crystallise  out 
side   by  side  in   unchanging  proportions.) 
We   now   consider  what   happens  when   a 
small  quantity  of  solute  is  separated  from 
or  taken  up  by  the  saturated  solution  by 
reversible    processes.      Let   the    saturated 
solution,  with  excess  of  solute,  be  placed  in 
a  cylinder  closed  below  by  a  semipermeable 
septum,  and  the  whole  immersed  in  pure 
solvent.     The  system  is  in  equilibrium  if  a 
Vl{i-  (il-  pressure  P,  equal  to  the  osmotic  pressure  of 

the  saturated  solution  when  the  free  surface  of  the  pure  solvent 
is  under  atmospheric  pressure,  is  applied  to  the  solution.  Dis- 
solution or  precipitation  of  solute  can  now  be  brought  about  by 
an  infinitesimal  decrease  or  increase  of  the  external  pressure, 
and  the  processes  are  therefore  reversible.  If  the  infinitesimal 
pressure  difference  is  maintained,  and  the  process  conducted  so 
slowly  that  all  changes  are  isothermal,  the  heat  absorbed  when 
a  mol  of  solute  passes  into  a  solution  kept  always  infinitely 


ELEMENTARY  THEORY  OF  DILUTE   SOLUTIONS    303 

near  saturation,  whilst  at  the  same  time  the  maximum  amount 
of  osmotic  work  is  done,  is  called  the  latent  heat  of  reversible 
dissolution  in  a  saturated  solution,  A'. 

If  the  changes  of  volume  are  executed  very  rapidly,  they  may 
be  made  adiabatic,  and  a  Carnot's  cycle  may  be  performed  with 
the  apparatus.  We  take  V,  the  total  volume  of  the  system  in  the 
cylinder,  as  the  abscissa,  and  P,  the  osmotic  pressure,  as  ordinate. 
Let 

V  =  increase  of  volume  which  occurs  when  one  mol  of  solute  is 
dissolved  by  entering  solvent  to  produce  saturated  solution,  at 
constant  temperature  T  ; 

A'  =  the  heat  absorbed  during  this  process. 

The  isotherms  T,  and  T  —  ST  are  parallel  to  the  Y  axis,  and  the 
whole  cycle  is  exactly  similar  to  that  investigated  in  connexion 
with  change  of  state.  The  reader  will  easily  prove  that : 

^  -  -  (1) 

dT  ~  TV  ' 

4m  need  not  be  written  (^™J    ,  since  P  is  a  function  of  T  alone 

provided  the  solution  always  remains  saturated,  further  increase 
of  volume,  i.e.,  entrance  of  solvent,  simply  increasing  the  amount 
of  solution  without  altering  its  concentration. 

Equation  (1)  was  deduced,  independently,  by  Le  Chatelier  (1885) 
and  by  van't  Hoff  (1886);  it  applies  generally  to  all  solutions, 
whether  concentrated  or  dilute. 

If  the  solution  is  dilute  (which  restricts  the  theory  to  sparingly 
soluble  substances)  we  have  : 

/V  =  RT  .         .         .        .     (2) 

But  1/V  =  £,  the  volumetric  molecular  concentration, 

.-.  P  =  £RT        .         .        .         .    (3) 

If  the  solution  had  been  prepared  by  simply  mixing  solvent  and 
solute  in  a  calorimeter,  without  any  performance  of  external 
osmotic  work,  the  heat  A  would  have  been  absorbed,  where 

A  =  A'  -  PY  =  A'  —  RT  .  .  .  (4) 
A  is  called  the  calorimetric  heat  of  solution  in  a  saturated 
solution. 

From  (1),  (3),  and  (4)  we  get : 

*  =  BT-**  (5) 

If  the  calorimetric  heat   of    solution  in  a  saturated  solution 


304  THERMODYNAMICS 


is  |  Dative'  *n  w^c^  case  tne  addition  of  a  small  quantity 
of  solute  to  a  large  volume  of  solution  infinitely  near  saturation 
gives  rise  to  j  g^^11  of  heat>  the  effect  of  rise  of  temperature 


«>*>*»*• 


This  is  an  example  of  the  application  of  a  very  general  theorem, 
formulated  somewhat  imperfectly  by  Maupertius,  and  called  the 
Principle  of  Least  Action.  We  can  state  it  in  the  form  that,  if 
the  system  is  in  stable  equilibrium,  and  if  anything  is  done 
so  as  to  alter  this  state,  then  something  occurs  in  the  system 
itself  which  tends  to  resist  the  change,  by  partially  annulling  the 
action  imposed  on  the  system. 

In  particular,  if  we  raise  the  temperature,  there  will  occur  some 
change  which  will  give  rise  to  absorption  of  heat.  If  a  saturated 
solution  of  potassium  nitrate  is  in  equilibrium  with  crystals  of 
solid  salt  at  a  particular  temperature,  and  if  we  now  raise  the  tem- 
perature, a  change  must  occur  which  absorbs  heat  and  so  tends  to 
cool  the  system.  This  is  the  dissolution  of  more  solid,  because 
the  heat  of  solution  is  positive,  that  is,  heat  is  absorbed  when  salt 
goes  into  solution.  But  if  we  have  a  saturated  solution  of  calcium 
sulphate,  there  will  occur  a  precipitation  of  solid  on  warming, 
because  the  heat  of  solution  is  negative,  and  heat  is  absorbed 
when  salt  comes  out  of  solution. 

The  principle  has  been  enunciated,  more  especially  in  con- 
nexion with  chemical  reactions,  by  van't  Hoff,  under  the  name  of 
the  Principle  of  Mobile  Equilibrium,  and  by  Le  Chatelier,  as  the 
Principle  of  Reaction. 

In  the  integration  of  (5)  we  must  know  A  as  a  function  of 
temperature : 

/         \ 

(6) 

Case  1. — If  the  interval  of  temperature  in  which  the  change  of 
solubility  is  required  is  small,  A  may  be  without  sensible  error 
assumed  to  be  constant : 

*dT  A 

=  ~~  RT  ~*~ C0ns  '        '     ^ 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS    305 

It  is  evident  that,  although  A  may  be  deduced  from  a  know 
ledge  of  the  solubility  curve  : 

A  =XT) 
we  cannot  deduce  the  solubility  from  a  knowledge  of  A. 

R/II  |? 
A  =  -       — *LT-  ...  (8) 


tf, 

Example. — Solubilities  of  succinic  acid  in  water : 
T!  =  273 
To  =  273  +  8-5 
£2  _  4-22 

fi  ~"  2-88 

j.  cal. 


R  =  1-985    , 

degree. 

1-985  X  log  ^||  X  2-3026 

=  6857  cal. 


fj M 

V273       281-5/ 


(A  obs.  =  6700  cal.). 

Equation  (8)  was  found  to  give  results  in  good  agreement  with 
experiment  by  van't  Hoff  (1885),  van  Deventer  and  van  der 
Stadt  (1892),  and  Noyes  and  Sammet  (1903).  An  agreement 
between  the  observed  and  calculated  values  of  A  implies  that  the 
solute  has  a  normal  molecular  weight  in  solution. 

Case  2.  —  If  equation  (6)  is  to  be  integrated  over  a  moderate 
range  of  temperature,  A  cannot  be  considered  as  constant. 

Suppose  that  s  grams  of  solute  saturate  100  grams  of  solvent, 
and  let  c1}  c2,  c  be  the  specific  heats  of  solute,  solvent,  and  solution, 
respectively;  then  if  Q'  is  the  heat  absorbed  in  the  process, 
Kirchhoffs  theorem  (§  58)  shows  that 

!/T~  =  (S  +  10°)C  ~  ^Tl  +  10°C'2)     '         '     (9) 
The  heat  of  solution  may  increase,  remain  constant,  or  decrease, 
with  rise  of  temperature,  according  as 


(s  +  100)c  =  (*C! 

< 
In  all  cases  investigated,  the  upper  inequality  obtains. 


306  THERMODYNAMICS 

If  I  is  the  heat  absorbed  when  1  gram  of  salt  forms  a  saturated 
solution, 

Q'  =  8l 

.  <a_  =  100  +  s  c  _  8d  +  iQQca  (10) 

c'i,  c2,  c  may  be  assumed  to  be  independent  of  temperature, 
_          100  4-  g  cT  _  sci  +  100c2  m 
s  s 

...  1  =  /0  +  (c  -  Cl)T  +  10°IC  ~^>  T  .         .         .     (11) 

Now  it  has  been  shown  that  the  difference  of  the  specific  heats 
of  the  solution  and  solvent  is  proportional  to  the  concentration, 
in  the  case  of  aqueous  solutions  (§  10).  Assuming,  therefore,  in 
general 

c  =  c2  +  ks        .        .        .        .     (12) 
where  k  may  be  positive  or  negative,  we  have 

I  =  10  +  aiT  +  aaT  =  lo  +  aT        .         .     (13) 
where  a\  =  c  —  ci, 

a2  =  (c  —  c%)/s,  are  constants. 

It  follows  from  what  has  been  said  that  a  is  negative.  For  a 
molecular  weight  of  solute,  we  have 

A  =  ml  =  Ao  —  aT         .         .         .     (14) 
where  a  =  —  ma,  is  a  constant. 
We  now  substitute  in  (6)  : 

A  n 

•  ^Jp  —  .5 /nT  +  const.      .         .     (15) 
From  (13)  we  have  : 

RT  R 

or  logs  =  A-?-  Clog  T        .        .        .        .'        .     (17) 

where  B  =  —  ^  X  2'3026,  C  =  m^ai  ^  ^   X   2'3026, 

A  =:  const.' 

The  analogy  between  (17)  and  the  Kirchhoff  vapour-pressure 
equation  (§  88)  is  evident.  R.  T.  Hardman  and  the  author  have 
shown  that  (17)  enables  one  to  calculate  the  solubility  when  the 
"  solubility  parameters  "  A,  B,  C  have  been  obtained,  and  this 


ELEMENTARY  THEOEY  OF   DILUTE    SOLUTIONS     307 

even  with    very  concentrated   solutions.      Thus,  with   aqueous 
solutions  of  cane  sugar  : 

A  =  —      32-285 

B  =  -  1283-65 

C  =  —      12-2267 


T 

sobs. 

s  calc. 

s  calc.  —  s  obs. 

273 

179-2 

179-2 

283 

190-5 

189-8 

-0-7 

293 

203-9 

203-2 

-0-7 

303 

219-5 

219-5 

— 

313 

238-1 

238-9 

+  1-7 

323 

260-4 

262-0 

+  1-6 

333 

287-3 

288-4 

+  1-1 

343 

320-5 

320-5 

— 

It  is  evident  that  the  laws  of  dilute  solution,  assumed  in  the 
deduction  of  (17),  cannot  apply  to  solutions,  or  rather  syrups, 
containing  more  than  three  times  as  much  sugar  as  water.  The 
analogy  between  (17),  and  KirchhofFs  vapour-pressure  equation  is 
therefore  surprisingly  extensive. 

A.  Findlay  (1902)  found  that  Eamsay  and  Young's  rule  for 
the  vapour  pressures  of  pure  liquids  (§  89)  has  an  analogue  in  the 
case  of  solutions.  If  TA,  TA '  are  two  temperatures  at  which  the 
substance  A  has  the  solubilities  s,  s' ,  and  TB,  TB'  two  tempera- 
tures at  which  another  substance  has  the  same  solubilities  in  the 
given  solvent  then  : 

T   '         T 

^  =  ^  +  C(TB'  -  TB) 

AB  ^B 

where  c  is  a  constant.     The  rule  is,  however,  not  very  closely 
followed. 

Le  Chatelier  (1888)  has  discussed  the  general  form  of  the 
solubility  curve  in  the  light  of  equation  (5).  If  dX/dT  is  negative 
(which  is  usually  the  case)  the  curve  begins  asymptotically  to  the 
T  axis,  and  is  convex  to  it.  It  then  passes  through  a  point 
of  inflexion,  and  is  concave  up  to  the  maximum  where 
A  =  0,  d^/dT  =  0.  If  A  then  becomes  negative,  the  solubility 

x  2 


308  THERMODYNAMICS 

decreases  again  with  rise  of  temperature.    The  greater  part  of  the 
theoretical  curve  has  been  realised  with  calcium  sulphate. 

The  equation  shows  that  the  solubility  curve  must  be  continuous  ; 
all  breaks  indicate  that  the  solid  phase  in  contact  with  the 
saturated  solution  has  altered  in  character,  and  we  really  have  to 
do  with  two  distinct  solubility  curves  meeting  at  an  angle.  This 
occurs,  for  example,  with  Glauber's  salt  at  32°'6,  for  this  is  the 
transition  temperature  for  the  reaction 

Na2S04  .  10H20  ^  Na2S04  +  10H20. 

The  curve  below  32°'6  is  the  solubility  curve  of  Na2S04.  10H20  ; 
that  above  32°'6  is  the  solubility  curve  of  Na2S04.  The  idea  that 
such  breaks  correspond  with  changes  of  "  hydration  "  in  the 
solution  is  quite  unfounded,  because  all  the  properties  of  the 
homogeneous  solution  pass  continuously  through  the  transition 
temperature. 

Case  3.  —  The  solute  changes  its  molecular  state  when  the  con- 
centration of  the  solution  is  altered. 

Let  us  suppose  that  instead  of  the  equation  : 


which"  holds  for  a  mol  of  solute  in  a  volume  V,  or  £  mols  of  solute 
in  unit  volume,  we  have  : 

P  =  5^  =  #BT       ....     (18) 

where  i  is  van't  HofF  s  factor,  and  is  a  function  of  concentration 
and  temperature.     Then  : 

x  =  A'  —  PV  =  A'  —  I.BT  .      .      .   (19) 


BT»  ,.     ',.        .     (20) 

If  the  range  of  temperature  is  very  small,  we  can  integrate  (20) 
on  the  assumption  that  i  is  constant : 

lnt=-fifi  +  const (21) 

But  if  the  range  is  at  all  considerable,  we  can  no  longer  regard 
i  as  independent  of  temperature,  in  other  words  we  must  take 


ELEMENTARY  THEORY    OF  DILUTE   SOLUTIONS    309 

account  of  the   alteration    of    the    extent    of    ionisation    with 
temperature. 

It  is  shown  later  that  i  and  £  are  related  by  an  equation : 

=  K (22) 


1  — 


where  a  =  i  —  1,  for  a  binary  electrolyte, 

K  =  a  constant  for  a  particular  temperature. 
(22)  is  called  Ostwald's  Dilution  Law. 
Now  (20)  can  be  written  : 


and  if  we  substitute  from  (22)  we  find  : 

\_  2      #        a(l-a)dlnK 

RT2  ~  2  -  a  dT  ~*      2  -  a      dT 

an  equation  deduced  by  Noyes  and  Sammet  (1903).  Examples  : 
o-nitrobenzoic  acid  :  A  calc.  by  (6)  =  6,480  cal.  A  obs.  =  6,025  cal. 
Potassium  perchlorate  :  „  „  =  12,270  cal.  „  =  1,213  cal. 
Similar  equations  may  of  course  be  deduced  for  a  polymerised 
solute  ;  in  this  case  i  <  1  in  (18). 

An  interesting  calculation  due  to  J.  Meyer  (1911)  enables  the 
transition  temperature  of  two  forms  of  a  substance  to  be  derived 
from  two  measurements  of  the  proportions  of  the  forms  in  the 
saturated  solution  at  two  temperatures.  If  accented  symbols  are 
used  for  one  form,  and  unaccented  for  the  other,  we  have  : 

A  =  Kin  &    ™>     and  A'  =  R/»  %  ^\ 
£1  la—  li  Ci  la  —  li 


.-.  A  —  A'  -  Q(heat  of  transition)  =  R/w      1,          2'    . 

CiCa    la  —  li 

At  the  transition  temperature  3,  both  forms  have  the  same 
solubility, 


Cl     *  —  -M 

the  ratio  of  the  concentrations  being  independent  of  the  solvent. 

f  '      T  •& 

At  the  second  temperature,  Q  =  Eln  f2-       2  ^ 

€2   *  —  ^2 

6.          rp  rp  ''^Cl  %2 

.  .    -J  —  lila  — 


310  THERMODYNAMICS 

134.    Heats  of  Solution  and  Dilution. 

The  exact  significance  of  A  in  the  equations  of  the  preceding 
section  must  be  remembered.  Different  "  heats  of  solution  "  have 
been  used,  and  among  these  we  have  : 

(1)  The  Differential  Heat  of  Solution,  L. 

(2)  The  Integral  Heat  of  Solution,  A. 

(3)  The  Reversible  Heat  of  Solution  A',  in  a  saturated  solution. 
Let  us  consider  a  mass  m  of  solid  solute,  and  a  mass  M  of  sol- 

vent, brought  together  in  a  calorimeter.  When  the  whole  has 
passed  into  a  homogeneous  solution  at  the  original  temperature, 
a  quantity  of  heat  Q  will  have  been  absorbed.  We  now  set,  by  way 
of  definition  : 


A(w»,M>,T)  =       .         .         .         .     (i) 

Let  us  now  consider  what  happens  when,  during  the  above  pro- 
cess, the  system  contains  a  mass  M  of  solvent  and  a  mass  p  of 
solute  and  a  further  small  quantity  dp  of  the  latter  goes  into 
solution. 

The  concentration  is  o-  •=.  ~,  and  if  BO  is  the  element  of  heat 

M 

absorbed  we  put,  by  way  of  definition  : 

SQ  =  L(<r,p,T)<1p 

=  LMdo-         ....     (2) 

If  the  whole  process  is  supposed  to  be  divided  up  into  an 
exceedingly  large  number  of  small  processes  of  this  kind,  in  the 
execution  of  which  the  concentration  of  the  solution  varies  from 

0  to  8  =  ~,  and  in  each  of  which  a  mass  dp  of  solute  goes  into 

solution,  we  have  : 

wA  = 


or  mA  =     L(<r,2),T)dp  =  M    L(a,p,T)d<r 


=  -  |  L(«rj),T)rf(r 


/.   A(«,p,T)  =  -s  I  L(crj), 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS     311 


or  L(s,p,T)  =      [*A(«,p,T)]  =  A  +  ,          .         .         .     (4) 

whence   the    names   "  differential  "    and   "  integral  "   heats   of 
solution. 

If  m  is  one  mol,  and  the  final  solution  is  saturated  : 

AQ»,T)  =  AfoT)         ....     (5) 
and  the  effect  of  pressure  may  be  neglected  in  most  cases. 

As  M  is  increased  in  comparison  with  m,  the  heat  of  solution 
approaches  a  limiting  value,  which  is  evidently  a  special  case  of 
the  differential  as  well  as  of  the  integral  heat  of  solution;  it 
represents  the  first  stage  in  the  supposed  series  of  small  processes 
when  the  solute  dissolves  in  initially  pure  solvent,  and  is  called 
the  licat  of  solution  at  infinite  dilution: 

L00(p,T)  =  L(0,j>,T)         .         .         .    (6) 

If  the  integral  heat  of  solution  is  independent  of  concentration, 
i.e.,  the  same  amount  of  heat  is  absorbed  when  unit  mass  of 
solute  dissolves  in  any  quantity  of  solute  : 

\(s,p,T)  =  L(s,p,T)       .        .        .     (7) 

A(s,^,T)  may  be  determined  directly  by  calorinietry  ;  if  its 
dependence  on  s  is  found  by  carrying  out  the  process  with 
different  values  of  s,  ~L(s,p,T)  may  be  obtained  by  differentiation. 

A  simple  graphical  method  of  effecting  the  calculation  is 
described  by  B.  Roozeboom  (Heterogcn. 
Gleichgeic.  II.).  We  take  AB  as  unit  length 
on  the  axis  of  abscissae,  and  let  the  point  a, 
where  Ba  =  x,  A«  =  1  —  x,  denote  the 
composition  of  a  mixture  of  x  parts  of  B 
with  (1  —  x)  parts  of  A.  The  corresponding 
ordinate  denotes  the  heat  absorbed  (positive 
or  negative)  in  the  formation  of  the  mixture.  A 
The  latter  will  not  usually  change  sign  with 
change  of  concentration  of  the  mixture, 
although  cases  in  which  this  occurs  (e.g., 
cupric  chloride,  the  hydrates  of  ferric  chloride, 
and  trichloracetic  acid,  in  water)  are  known. 
The  summits  of  the  ordinates  will  therefore  lie  on  a  continuous 
curve,  either  wholly  above  (Q  >  0),  or  wholly  below  (Q  <  0)  the  com- 
position axis,  as  for  example,  A6B  or  A/3B.  ab  =  heat  absorbed  in 
the  formation  of  unit  mass  of  the  mixture  containing  Ba  =  x  parts 


312  THERMODYNAMICS 

of  B.   The  heat  for  1  part  of  B  is  -  times  this  =  BC.    If  x  becomes 

greater  and  greater,  the  point  C  moves  upwards,  and  finally  attains 
the  limiting  position  F.  BF  represents  the  heat  for  solution  of 
1  part  of  B  in  an  infinite  amount  of  solvent,  i.e.,  the  differential 
heat  of  solution  for  infinite  dilution.  BC  is  the  integral  heat  of 
solution.  Similarly,  if  IG  is  drawn  tangent  to  the  curve,  BG 
represents  the  differential  heat  of  solution  of  1  part  of  B  in  an 
infinite  amount  of  the  solution  represented  by  a. 

If  a  solution  of  concentration  s,  containing  unit  mass  of 
dissolved  substance,  is  mixed  with  an  infinite  amount  of  pure 
solvent,  the  heat  absorbed  is  called  the  integral  heat  of  dilution, 
A(s,T,p).  The  reason  for  the  use  of  an  infinitely  large  amount 
of  solvent  is  that  A  depends  on  *,  but  after  the  dilution  has  pro- 
gressed to  a  greater  or  less  degree,  depending  on  the  character  of 
the  solute,  any  further  addition  of  solvent  gives  rise  to  no  further 
heat  effect,  or  at  least  one  which  is  altogether  too  small  to 
detect. 

The  heat  absorbed  when  unit  mass  of  solute  is  dissolved  in  an 
infinite  amount  of  solvent  is  the  differential  heat  of  solution  for 
zero  concentration,  LO,  and  this  is  evidently  equal  to  the  integral 
heat  of  solution  for  concentration  s  plus  the  integral  heat  of 
dilution  for  concentration  s  : 

Lo  =  A.  +  Ag     ,         .         .         .          .       (8) 

.,^-+^•  =  ^0    =0          .         .         .       (9) 

3s         cs         ds 

.'.  from  (4) 

A,  =  L.-«^        .        .        .        .        .    (10) 

Also  ^  =  rs-r    .       .       .       .       .       .    (ii) 

where  F,,  F  are  the  total  heat  capacities  of  the  solution  and  of  its 

,  8L,       dAs   .      a2A,  . 
components,  and  ~  =  ^  +  s  -^^  from  (4) 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS    313 

135.   The  Distribution  Law. 

If  twopartially  miscible  liquids  are  shaken  together,  two  saturated 
solutions,  one  of  A  in  B,  and  the  other  of  B  in  A,  will  in  general 
result.  If  now  a  third  substance  is  added,  which  dissolves  in  either 
pure  liquid,  it  will  go  into  solution  in  both  layers.  Thus,  if 
ether  and  water  are  agitated  together,  the  lower  layer  will  consist 
mainly  of  water,  and  the  upper  mainly  of  ether.  If  iodine  is  now 
added,  it  will  nearly  all  go  to  the  ethereal  layer,  but  a  little 
dissolves  in  the  aqueous  layer,  as  is  evident  from  the  colour. 

It  was  experimentally  established  by  Berthelot  and  Jungfleisch 
(187*2),  that :  "  a  body  brought  in  contact  with  two  liquids,  in 
each  of  which  it  is  soluble,  always  divides  itself  between  them  in 
a  simple  ratio,  however  great  maybe  its  solubility  in  one  of  them, 
and  the  excess  of  the  volume  of  this  same  solvent.  The  quantities 
dissolved  simultaneously  by  the  two  liquids  stand  to  one  another 
in  a  constant  ratio  which  is  independent  of  the  relative  volumes 
of  the  two  liquids."  This  ratio  is  called  the  coefficient  of  distribu- 
tion, or  the  partition  coefficient,  k. 

Thus,  succinic  acid  is  shared  between  ether  and  water  with 
the  following  values  of  /,- : 


0-024  0-0046  5-2 

0-070  0-013  5-3 

0-121  0-022  5-5 


TI,  c-2  are  the  grams  of  solute  in  10  c.c.  water  and  ether  respectively. 

It  was  also  found  experimentally  that  two  solutes  were  dis- 
tributed between  a  pair  of  solvents  as  if  each  were  present  alone. 
(This  is  analogous  to  Dalton's  law  of  partial  pressures.) 

Berthelot  and  Jungfleisch  considered,  from  their  experimental 
results,  that  for  any  given  trio  of  substances,  A-  depended  on  the 
temperature,  and  on  the  concentration  of  the  shared  substance. 
It  was  shown,  however,  on  theoretical  grounds,  by  Aulich,  and 
Xernst,  that,  provided  the  solute  does  not  alter  its  molecular 
state  in  passing  from  one  solvent  to  another,  A-  should  be  indepen- 
dent of  concentration,  but  will  depend  on  the  temperature. 


314 


THERMODYNAMICS 


We  assume  that  an  equilibrium  can  subsist  with  a  specified 
pair  of  solvents  in  contact  and  a  solute  distributed  between  them 
in  any  one  ratio  of  concentrations  : 


We  have  now  to  prove  that,  if  £1  is  increased  by  d£i,  then 
will  increase  by  <7£2,  such  that  : 


This  will  establish  the  law  for  all  concentrations  within  the  range 

to  which  the  laws  of  dilute  solutions  apply,  because  the  same 

argument  could  be  used  with  the  second  pair  of  concentrations, 

fi  -f-  dfi,  and  £2  +  ^£2,  as  initial  concentrations,  and  so  on,  hence 

£!/£2  =  constant  =  k        .         .         .         (a) 

This  means  that  if  one  of  the  magnitudes  £1  or  £2  has  been 
fixed  by  arbitrary  choice,  the  other  assumes  the  value  given  by  (a), 
A-  being  a  fixed  constant  for  all  values  of  £. 

Let  there  be  two  cylinders  containing  the  pairs  of  solvents  in 


FIG.  63. 

which  the  same  solute  exists  under  the  osmotic  pressures  PI,  P2 
in  the  first,  PI  +  <^Pi>  ?2  +  ^P2  in  the  second,  and  let  Vi,  V2  and 
Vi  —  rfVi,  Va  —  rfVa  be  the  corresponding  volumes  of  solution  con- 
taining a  mol  of  solute.  The  suffixes  refer  to  the  first  and  second 
solvent,  respectively. 


ELEMENTAEY  THEORY  OF  DILUTE   SOLUTIONS     815 

We  now  press  in  the  piston  1  so  that  a  mol  of  solute  goes  out 
of  the  Yi  space  into  the  Y2  space  across  the  plane  of  contact, 
at  the  same  time  allowing  the  piston  2  to  move  out*so  that  the 
requisite  volume  of  solvent  enters  and  the  concentrations  are 
unaltered.  The  work  done  is 

d]=--PxVi  +  P,V«. 

That  volume  of  the  second  solution  which  contains  a  mol  of 
solute,  viz.,  V2,  is  now  isolated,  and  the  volume  f/V2  of  solvent 
removed  through  the  semipermeable  piston.  The  work  done  is 

[2]  =  -  P2r/V2, 

The  resulting  solution  is  now  mixed  with  the  identical  solution 
in  the  other  cylinder,  and  the  mol  of  solute  sent  into  the 
YI  —  d\ri  space,  whereby  the  work 

[3]  =  (P!  +  rfPO  (Vi  -  dVO  -  (Pa  +  dPa)  (Y2  -  <fVa) 
is  done. 

Lastly,  the  concentration  is  changed  to  Vi,  the  work 

[4]=P,iJV, 

being  done,  and  the  resulting  solution  is  returned  to  the  initial 
compartment. 

The  isothermal  and  reversible  cycle  is  now  completed,  hence 
[1]  +  [2]  +  [3]  +  [4]  =  0 
.-.   Vir/Pi  =  V*/Pa          .          .          .          .      (1) 

-prp 

But  P  =  ^=£RT        .        .        .        .     (;2) 

/.  r/P  =  RT^  (T  const.)          .         .         .     (3) 
hence  ±-       =  ......      4 


The  effect  of  the  addition  of  a  third  substance  to  one  of  two 
partially  miscible  liquids  on  the  solubility  relations  of  the  two 
liquids  has  been  considered  by  Nernst  (1890).  Let  us  suppose 
that  ether  and  water  are  taken,  and  that  s  denotes  the  solubility 
of  pure  ether  in  water  at  the  temperature  T.  Now  let  a  third 
substance  which  dissolves  only  in  the  ether  be  added  to  that 
solvent  (say  stearic  acid),  and  let  s'  be  the  solubility  of  ether  con- 
taining this  substance.  It  is  obvious  that  the  solubility  of  the 
ether  in  water  must  be  equal  to  that  of  ether  vapour,  and  the 
latter  may  be  calculated  (with  the  usual  restrictions)  from 
Henry's  law  :  * 

s  :  s'  =  p  :  p' 


316  THERMODYNAMICS 

where  p,  p'  are  the  partial  pressures  of  ether  vapour.     But,  from 
the  equation 

«-  ,    p       n 

In  ±-f  =  — 

P        »o 

,  n        ,    p        ,     s         s  —  s' 

we  have  —  =  In2-^  =  In  —,  =  -  -.  —  very  nearly, 

»o          p  s  s' 


.      .      .  , 

s  HO  +  n 

so  that  the  relative  lowering  of  solubility  is  proportional  to  the 
number  of  mols  of  dissolved  substance  divided  by  the  number  of 
mols  of  solvent  and  solute. 

The  freezing-point  of  water  in  contact  with  ether  is  lowered  by 
3°'85  owing  to  the  amount  of  the  latter  dissolved.  If  now  a  third 
substance  (e.g.,  benzene)  is  added  to  the  ether,  which  does  not 
dissolve  in  the  water,  the  freezing-point  will  be  raised  on  account 
of  the  diminished  solubility  of  the  'ether.  The  diminution  of 
solubility  of  the  ether  is  quite  apparent  to  the  eye  if  the  liquids 
are  contained  in  a  graduated  tube.  •;:»(,;*•.•*• 

If  the  substance  shared  between  two  solvents  can  exist  in 
different  molecular  states  in  them,  the  simple  distribution  law  is 
no  longer  valid.  The  experiments  of  Berthelot  and  Jungfleisch, 
and  the  thermodynamic  deduction  show,  however,  that  the 
distribution  law  holds  for  each  molecular  state  separately.  Thus, 
if  benzoic  acid  is  shared  between  water  and  benzene,  the 
partition  coefficient  is  not  constant  for  all  concentrations,  but 
diminishes  with  increasing  concentration  in  the  aqueous  layer. 
This  is  a  consequence  of  the  existence  of  the  acid  in  benzene 
chiefly  as  double  molecules  (C6H5COOH)2,  and  if  the  amount  of 
unpolymerised  acid  is  calculated  by  the  law  of  mass  action  (see 
Chapter  XIII.)  it  is  found  to  be  in  a  constant  ratio  to  that  in  the 
aqueous  layer,  independently  of  the  concentration  (cf.  Nernst, 
Theoretical  Chemistry,  2nd  Eng.  trans.,  486  ;  Die  Vertcilungssatz, 
W.  Hertz,  Ahrens  Sammlung,  Stuttgart,  1909). 

136.     Effect    of  Pressure   on   Solubility. 

If  a  substance  dissolves  with   (increase  Of  total  volume,  the 

(decrease 


principle  of  Le  Chatelier  shows  that  the  solubility  is 

by  an  increase  of  the  total  pressure  on  the  system  (Sorby,  1863). 


ELEMENTARY  THEORY   OF   DILUTE   SOLUTIONS     317 

The  magnitude  of  the  change  was  calculated,  independently, 
by  Guldberg  (1870),  and  by  Braim  (1887). 

Let  us  suppose  the  apparatus  described  in  §  133  is  enclosed  in 
a  vessel  into  which  an  indifferent  gas  may  be  pumped,  so  that 
the  whole  system  can  be  exposed  to  any  total  external  pressure 
desired.  If  there  is  a  change  of  total  volume  when  solute  passes 
into  solution,  it  will  give  rise  to  the  performance  of  work  by,  or 
against,  the  total  pressure,  quite  independently  of  the  osmotic 
work  derivable  from  the  apparatus  inside. 

Suppose  the  osmotic  pressure  under  total  pressure  p  is  P,  and 
let  Arp  be  the  change  of  total  volume  occurring  when  a  mol  of 
solute  is  dissolved  under  total  pressure  p.  Let  V,,  be  the  change 
of  volume  of  the  system  inside  the  osmotic  apparatus  when 
solvent  is  admitted  so  as  to  dissolve  a  rnol  of  solute  (cf.  §  133). 

P  -f-  SP,  Ar.,  ,  6/,,  V,,  +  sp  are  the  magnitudes  corresponding 
to  a  total  external  pressure  p  -\-  fy>. 

Now  let  the  following  reversible  isothermal  cycle  be  per- 
formed : 

(1)  Increase  p  to  p  +  §p. 

(2)  Allow   solvent   to  enter  until    a  mol  of  solute  has  been 
dissolved  inside  the  osmotic  apparatus.     The  work  done  is 


(3)  Decrease  the  external  pressure  to  p. 

(4)  Express  solvent  until  a  mol  of  solute  is  precipitated  from 
the  saturated  solution,  and  the  work 

-  PV,  -  pto, 

is  done. 

Everything  is  now  in  the  initial  state,  and  the  total  work  done 
must  vanish.  By  reason  of  the  slight  compressibility  of  the 
solution  and  solvent,  the  amounts  of  work  done  in  (1)  and  (3) 
may  be  taken  as  equal  and  opposite. 

/.  (P  +  8P)V,  +  *,  +  (  p  +  S;>)A0)  +  S]>  -  PV,  -  7>Arp  =  0     .     (1) 
But  V^  =  Y,+^, 

and  Ar,  +  Sp  —  Arp  -f-  -g—  "  8p, 

whence,  if  we  neglect  small  quantities  of  the  second  order, 

0    ....     (2) 


318  THERMODYNAMICS 

Now  1/Vp  =  £p,  the  volumetric  molecular  concentration, 

TfP 
and  8P  =  |r^ 

i    ap  ...      ap, 


OT-I-  -m  •  •  •-  •  •  • <8) 

or  ~-  = ^J5    .  .  .  ...  (3«) 

"1  ~\T 

Now  (§133):  Vp=    ~           .  .  .  .         .  .     (4) 


ap 

a_T  _      TAg,  8f 

•"•  a^  ~   "  A'  "  •  ap  ~   "  A'  aT 

This  equation  is  general,  and  applies  to  solutions  of  all 
concentrations. 

From  equation  (3)  we  see  that,  since  ~  is  essentially  positive, 

-—is  of  opposite  sign  to  Ar  (Sorby's  rule). 

The  quantitative  relations  have  been  tested  by  Braun,  by 
Stackelberg  (1896),  and  by  Cohen,  Inouye,  and  Euwen  (1911). 

Ammonium  chloride  passes  into  solution  with  increase  of 
total  volume,  and  hence  its  solubility  should  be  diminished  by 
increase  of  pressure.  Sodium  chloride,  on  the  contrary,  dissolves 
with  contraction,  and  its  solubility  should  be  increased  by  rise  of 
pressure.  Above  1,530  atm.,  however,  the  latter  salt  dissolves 
with  expansion,  and  its  solubility  then  decreases  with  pressure. 
These  deductions  from  the  equation  have  been  confirmed. 

We  may  assume,  as  a  close  approximation,  that  the  osmotic 
pressure  is  defined  by  the  equation : 

P  =  £RT  ....     (6) 


ELEMENTARY  THEORY  OF  DILUTE    SOLUTIONS     319 

dln£ 
thence 


an  equation  due  to  Planck  (1897). 

It  will  be  noticed  that  in  the  deduction  of  (6)  it  was  assumed 
that  no  change  of  volume  occurred  when  solute  passed  into 
solution.  If  the  change  is  small,  however,  which  is  always  the 
case,  we  can  neglect  the  work  done  by  the  osmotic  pressure 
against  this  change  of  volume  in  comparison  with  that  done 
against  the  change  of  volume  V,,. 

If  the  solute  dissociates  with  increasing  dilution,  the  equation 
(7)  requires  modification  ;  thus,  van  Laar  (1893)  deduced  for  a 
binary  electrolyte  : 

AB=A+  +  B- 

the  equation  : 

Ar        2-q 


'         BT  '       2 

where  a  is  the  degree  of  electrolytic  dissociation  (cf.  §  133). 
This  is  readily  obtained  by  putting  : 

P  =  i£RT          ....     (6a) 
t  =  1  +  a        .         .         .         .       (9) 


and  substituting  in  (3). 

The  following  results  have  been  obtained  : 
NaCl  :  s  =  35-898  +  0'001647p  —  0'0000003286^2 

Mannitol  :     s  =  20'65    +  0'000931p  —  O'OOOOOOISOG/ 


(Cohen,  Inouye,  and  Euwen,  Zeitschr.  physik.   Chem.,  67,  432, 
1909;  75,257,1910.) 


137.     Other  Causes   Modifying  Solubility. 

Besides  the  effect  of  temperature  and  pressure,  the  mechanical 
pressure  exerted  on  the  solid  phase,  and  its  state  of  division, 
influence  (although  only  to  a  slight  extent)  its  solubility  in  the 
liquid.  Thus,  if  a  moist  precipitate  is  exposed  to  pressure  in  a 
filter-press,  it  usually  aggregates  together,  and  Hulett  (1901) 
showed  that  the  effect  of  division  (i.e.,  of  surface  tension)  becomes 


320 


THERMODYNAMICS 


appreciable  in  the  case  of  calcium  and  barium  sulphates  when  the 
diameter  of  the  grains  falls  below  1'9  /x(>  =  10~*crn.) 
CaS04  (ordinary)  at  25°  :  f  =  0-01533 
CaC04  grains   '  <  0"2  /x  :  £  =  0'01869. 

This  effect  appears  to  be  of  importance  in  the  case  of  normal 
galvanic  cells,  the  electromotive  forces  of  which  depend  on  the 
concentration  of  solutions  in  equilibrium  with  depolarising  solids 
such  as  calomel  or  mercurous  sulphate.  The  exact  relationships 
are,  unfortunately,  not  yet  wholly  elucidated. 

G.  Hulett,  Zeitschr.  physik.  Chem.,  37,  385,  1901 ;  Hulett  and 
Allen,  Joimt.  Amcr.,  Chem.  Soc.,  24,  667,  1902. 

138.     Dilute  Solid  Solutions. 

The  theory  of  the  depression  of  freezing-point  of  a  solvent  by 
addition  of  a  soluble  substance  considered  in  §  132  is  based  on 


FIG.  64. 

the  assumption  that  the  solid  separating  is  the  pure  frozen 
solvent.  In  this  case  the  equilibrium  temperature  is  a  function 
of  the  composition  of  the  liquid  phase,  but  obviously  not  of  the 
composition  of  the  solid,  since  the  latter  remains  invariable. 

In  certain  cases,  however,  the  solid  which  separates  is  a 
homogeneous  mixture  of  both  components,  and  hence  may  be 
referred  to  as  a  solid  solution.  These  are  often  called  "  mixed 
crystals,"  but  the  name  is  clearly  unsuitable  in  view  of  the 


ELEMENTARY  THEORY  OF   DILUTE    SOLUTIONS    321 

homogeneous   character   of   the   phase ;    a   mixed   crystal   is   a 
mixture  in  a  crystal,  not  a  mixture  of  crystals. 

The  effect  on  the  equilibrium  temperature  was  investigated  by 
van't  Hoff  (1890)  by  means  of  the  vapour-pressure  diagram. 
Instead  of  the  diagram  of  §  132,  in  which  the  vapour-pressure 
curves  of  the  solid  are  confined  to  a  single  curve  OA  for  the  pure 
frozen  solvent,  we  have  now  Fig.  64.  AB,  BC  are  the  curves 
for  pure  solid  and  liquid  solvent  respectively,  B(T0)  being  the 
freezing-point  of  pure  solvent.  B2C2  is  the  curve  for  a  liquid 
solution,  and  if  the  solid  does  not  dissolve  the  second  component, 
the  freezing-point  is  T2,  and  the  depression  T0  —  T2.  If,  how- 
ever, the  solid  also  dissolves  some  of  the  second  component, 
its  vapour  pressure  is  lowered,  and  the  curve  AiBi  is  obtained 
for  the  solid  phase  instead  of  AB.  The  freezing-point  is  now 
T3,  the  abscissa  of  the  intersection  of  the  curves  of  the  solid 
and  liquid.  The  depression  T0  —  T3  is  obviously  weaker  than 
before,  and  in  some  cases  the  freezing-point  may  be  actually 
raised,  the  intersection  lying  to  the  right  of  Tlt  as  at  T± 

If  we  assume  that  the  lowering  of  vapour  pressure  is  pro- 
portional  to  the   concentration   of   solute,   in   both   the   liquid 
and  solid  solutions,  and  that  the  solute  is  involatile,  then : 
Tj  -  T3  _  BB: 
T!  -  T2  ~  BB2 

This  applies  to  what  may  be  called  dilute  solid  solutions 
and  has  been  confirmed  by  Bijlert  (1891)  for  iodine,  and  for 
thiophene,  dissolved  in  benzene,  and  by  Bruni  for  iodoforin  in 
bromoform. 

The  form  of  the  curves  also  leads  to  an  important  rule  deduced 
by  Roozeboom.  It  is  obvious  that  the  abscissa  of  the  inter- 
section of  the  curves  of  the  solid  and  liquid  solutions  will  lie  to 
the  left  (T3)  or  to  the  right  (T4)  of  TI,  according  as  BB2  is  greater 
or  less  than  BBx  respectively.  These  lengths  will,  by  Raoult's 
law,  be  proportional  to  the  concentrations  of  the  solute  in  the 
liquid  and  solid  phases,  respectively,  and  hence  the  concentration 
of  that  component,  by  the  addition  of  which  the  freezing-point  is 
depressed,  will  be  greater  in  the  liquid  than  in  the  solid  phase, 
and  the  concentration  of  that  component,  by  addition  of  which 
the  freezing-point  is  raised,  will  be  greater  in  the  solid  than  in 
the  liquid  phase. 

(Bruni,  Feste  Losungen,  Ahrens  Sammlung. 


CHAPTEK  XII 

CHEMICAL    EQUILIBBIUM    IN    GASEOUS    SYSTEMS 

139.     Chemical    Equilibrium. 

If  vi,  r2,  .  .  i';  mols  of  the  substances  Alf  A2,  .  .  A,-  are  mixed 
together,  so  as  to  form  either  a  homogeneous  phase,  or  a  hetero- 
geneous system  of  two  or  more  phases,  the  system  may  behave 
in  one  of  two  ways  : 

(i.)  The  substances  disappear  as  such,  and  v± ,  v%  .  .  v?  mols 
of  the  new  substances  A/,  A2'  .  .  A/  appear  in  their  place.  The 
change  is  called  a  chemical  reaction,  and  may  proceed  either 
very  rapidly,  as  is  the  case  with  explosive  reactions,  or  reactions 
of  neutralisation  in  solution,  or  else  may  go  on  with  a  finite  and 
measureable  velocity.  The  determination  of  the  velocity  with 
which  a  reaction  progresses,  and  of  the  influence  of  various 
conditions,  such  as  the  concentration  of  the  reacting  substances, 
the  temperature,  and  catalysts,  on  the  velocity,  forms  the  subject 
of  that  branch  of  chemistry  called  chemical  kinetics,  or  chemical 
dynamics. 

(ii.)  The  initial  substances  persist  in  unchanging  amount  for 
an  indefinite  period  of  time,  so  that  the  composition  of  the 
system  is  independent  of  time.  The  system  is  then  said  to  be  in 
a  state  of  chemical  equilibrium,  and  the  complete  study  of  states 
of  chemical  equilibrium  is  the  aim  of  that  branch  of  chemistry 
which  is  called  chemical  statics. 

By  means  of  experimental  investigation,  states  of  chemical 
equilibrium  may  be  divided  into  two  groups.  For  this  purpose 
we  change  the  amounts  of  some  of  the  components  in  the  system, 
and  see  if  any  change  in  the  amounts  of  the  other  components 
ensues. 

(a)  If,  when  the  amount  of  any  one  component  is  changed  by 
any  quantity,  however  small,  a  corresponding  change  in  the 
amounts  of  one  or  more  of  the  other  components  ensues,  the 
state  is  said  to  be  one  of  true  chemical  equilibrium.  A  mixture 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS     323 

of    acetic     acid,    alcohol,    water,    and     acetic     ester,    in     the 
proportions : 

iCH3COOH  +  |C2H5OH  +  §CH3COOC.2H5  +  f  H20 
satisfies  this  condition  at  the  ordinary  temperature.  Berthelot 
and  Saint-Gilles  could  detect  no  change  in  the  composition  of 
such  a  mixture  after  it  had  stood  for  seventeen  years,  but  if  the 
slightest  change  in  the  amount  of  one  component  is  produced  by 
adding  a  little  of  it  to  the  mixture,  the  amounts  of  the  other 
components  very  soon  alter  accordingly. 

(b)  If,  when  the  relative  amounts  of  some  of  the  components 
are  changed,  no  change  in  the  proportions  of  the  rest  ensues,  the 
state  is  certainly  not  one  of  true  chemical  equilibrium.  In  this 
case  it  may  be  possible  to  get  the  reaction  to  proceed  by  the 
introduction  into  the  system  of  a  so-called  catalyst,  which 
remains  chemically  unchanged  after  the  reaction,  as  in  the  case 
of  the  catalytic  influence  of  platinum  sponge  on  a  mixture  of 
hydrogen  and  oxygen  gases ;  or  else  the  reaction  cannot  be  so 
instituted,  as  in  the  case  of  metallic  gold  and  oxygen.  The  first 
case  is  an  example  of  false  chemical  equilibrium,  the  second  of  a 
system  of  chemically  indifferent  substances.  It  may  be  that  these 
distinctions  are  only  arbitrary,  and  that  all  substances  may  really 
react  when  this  is  possible,  but  sometimes  the  reaction  is  either 
so  slow,  or  proceeds  to  such  a  limited  extent,  that  it  is  imper- 
ceptible. In  practice,  however,  there  is  usually  not  the  slightest 
difficulty  in  making  out  to  which  class  a  given  system  belongs. 
In  what  follows,  "  chemical  equilibrium  "  is  to  be  taken  as 
meaning  "  true  chemical  equilibrium." 

140.     Reversible  Reactions. 

If  acetic  acid  and  alcohol  are  mixed  in  equimolecular  proportions, 
a  reaction  ensues  leading  to  the  formation  of  an  equilibrium 
mixture  of  the  two  substances  with  the  ester  and  water.  If  water 
and  the  ester  are  mixed  in  equimolecular  proportions,  a  reaction 
ensues  leading  to  the  equilibrium  mixture  of  these  two  substances 
with  the  acid  and  alcohol.  Thus  the  reaction  : 

acid  +  alcohol  =  water  +  ester 

may  proceed  in  either  direction  ;  it  is  called  a  reversible  reaction, 
and  formulated : 

acid  +  alcohol  ^±  water  -\-  ester. 

T  2 


324  THERMODYNAMICS 

The  fact  that  the  same  equilibrium  mixture  is  attained  whether 
we  start  with  the  substances  on  the  left,  or  those  on  the  right, 
follows  from  the  condition  that  this  shall  be  a  state  of  true 
equilibrium.  For  if  two  equilibrium  mixtures,  say  a  and  a', 
could  result  in  the  two  cases,  one  would  of  necessity  contain  the 
components  in  proportions  different  from  those  in  the  other. 
The  one,  say  a',  which  contains  any  specified  component  in 
excess  over  the  other  mixture,  could  be  produced  from  the  latter 
by  adding  to  it  the  requisite  excess  of  that  component.  But 
if  a  is  a  state  of  true  equilibrium  this  will  necessarily  give  rise  to 
some  chemical  change  in  the  system,  and  hence  a'  cannot  be  a 
state  of  true  equilibrium  if  it  differs  at  all  from  a. 

In  general,  the  state  of  true  equilibrium  is  represented  by  the 
symbol : 

i'lAi  +  r2A2  +  .  .  ^±  i-i'Ai'  +  2-2' A2'  +  .  . 
where  r-t,  v{  are  the  numbers  of  mols  of  the  components  A(,  A/ 
taking  part  in  the  reversible  reaction. 

141.     Chemical  Equilibrium  in  Gaseous  Systems. 

The  thermodynamic  theory  of  equilibrium  was  first  stated,  in 
a  general  way,  by  Horstmann  in  1873  (cf.  §  50),  who  also 
obtained  explicit  equations  of  equilibrium  in  the  case  where  it  is 
established  in  a  gas,  and  showed  that  these  were  in  agreement 
with  the  data  available  at  that  time,  and  with  his  own 
experiments. 

Since  in  the  majority  of  cases  we  have  to  deal  with  constant 
pressure,  it  is  most  useful  to  express  the  conditions  of  equilibrium 
in  terms  of  the  potential  <f>. 

When  several  gases  are  mixed  together,  or  have  resulted  from 
chemical  change,  so  as  to  form  a  homogeneous  gas,  it  is  possible 
to  assign  to  the  system  a  total  potential  <£.  If  any  change  of  com- 
position occurs,  there  will  be  simultaneously  a  variation  of  <f>,  and 
if  the  changes  occur  reversibly  we  shall  assume  that  this  variation 
takes  place  continuously.  The  condition  of  equilibrium  at  a 
constant  temperature  and  pressure  can  then  be  expressed  in  the 
statement  that  all  possible  virtual  isothermal-isopiestic  changes 
in  composition  of  the  system  will,  when  the  latter  is  in 
equilibrium,  leave  <f>  unchanged  to  the  first  order : 

fy  =  0 (1) 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    325 

subject  to  the  conditions  : 

ty  =  0.        ....        ..      (2) 

5T  =  0  .....       (3) 

The  equilibrium  is  stable  if  <£  is  an  absolute  minimum  : 

.*.  S2<£>0         .         .         .         .-    (4) 

subject  to  the  conditions  (1),  (2),  and  (3). 
These  relations  are  perfectly  general. 

142.     Gas  Mixtures  with  Convertible  Components. 

A  mixture  of  gases  which  can  undergo  chemical  change  is 
distinguished  from  mixtures  of  inert  gases  (§  122)  by  the  name 
</«-?  mixture  icith  convertible  components  (Gibbs,  Scientific  Papers, 
I.  172). 

We  shall  assume  that  at  every  instant  the  mixture  can  be 
regarded  as  an  ideal  gas  mixture,  and  therefore  as  having  a 
potential  equal  to  the  sum  of  the  potentials  of  the  components 
present  at  that  instant,  each  occupying  the  whole  volume  of  the 
mixture  (§  122). 

In  this  case  the  total  potential  of  the  mixture  is  : 


R/n        +  R  +  R/WC.        .         .     (i) 

where  </,(T)  is  a  function  of  temperature,  the  exact  form  of  which 
is  (to  an  arbitrary  linear  function  of  temperature)  known  when 
the  molecular  heats  are  known  as  functions  of  temperature  : 


-joxr-Tp^ 


0,{T)  =  U(i'  -  TS'j'  +  I  C'jV/T  -  T  |  ^~  +  RTf«M;  .         .     (2) 

where  C<*  =  /(T) .     (3) 

Now  put  everything  in  the  bracket  of  (1),  except  Elnch  equal 
to  <p,. : 

9,  is  a  function  of  temperature  and  pressure  which  will  receive 
more  consideration  later ;  for  the  present  we  see  that  it  does  not 
depend  on  the  concentrations,  and  is  therefore  constant  during 
the  virtual  change  (1)  of  §  141. 

;»</)  =  TS»  ,(<?,  +  R/nCi)  .         .         ...         .     (5) 


=  T2(o{  -f 


326  THEEMODYNAMICS 

But  in  the  equilibrium  state : 
S</>  =  0 

.-.  T2(9s  +  R/wc,-)8»;  +  T2w38Znc-j  =  0         .         .     (P) 
Now  /??f  j  =  /??»;  —  Inn,  where  n  =  S»; 


f«.  $H  ^  fill 

-,  —  ^oii; z,n-  =  on n  =  0. 

n  n 

The  term  with  89;  vanishes,  since  9,-  does  not  alter  with  change 
of  composition. 

The  second  part  of  (6)  therefore  vanishes,  and  : 

This  equation  applies  to  nearly  all  practical  cases  of  gaseous 
equilibria,  and  it  may  be  called  the  Canonical  Potential  Equation 
for  Gaseous  Equilibrium. 

Corollary  1. — If,  for  any  gaseous  component,  we  have : 

b?i{  =  0     .         .        ,        .         .     (8) 

the  term  relating  to  that  component  vanishes  from  the  canonical 
equation,  and  we  have  to  take  account  of  the  presence  of  that 
component  only  in  estimating  the  concentrations  of  the  other 
gases.  Equation  (8)  shows  that  this  particular  gas  takes  no 
part  in  the  chemical  change,  and  we  shall  therefore  call  it  an 
inert  gas. 

Corollary  2. — Equation  (7)  shows  that  the  equilibrium  is 
independent  of  the  absolute  masses  of  the  components,  and 
depends  only  on  their  relative  amounts,  that  is,  on  their  concen- 
trations. Thus,  if  a  gas  mixture  in  equilibrium  is  divided  into 
two  or  more  parts  by  diaphragms,  these  parts  remain  in 
equilibrium. 

Example — If  the  specific  heats  are  constant,  show  that : 

T  "R  TT(i) 

9i  =  C',;'(l  -  ZwT)  —  R/w  -  -  S;"  -  R/H  ~  +  R  +  ~r  -       (9) 
P  M, 

Notice  that  this  can  be  split  into  three  parts : 

T 

(i.)  —  Eln  — 1-  R,  which  depends  solely  on  external  conditions. 

TJ<* 
(ii.)  —  Bi*  +  R/»M,  +    ~,  which  depends-  on  the  molecular 


CHEMICAL  EQUILIBEIUM  IN  GASEOUS  SYSTEMS     327 

weight,    the   initial   states    of    entropy   and    energy,    and    the 
temperature. 

(iii.)  Cl-'^l  —  /ttT),  which  is  a  function  of  temperature  alone. 

If,  besides  the  temperature,  the  total  volume  V  is  maintained 
constant,  instead  of  the  pressure,  the  equilibrium  condition  is 
that  the  free  energy  must  be  a  minimum  : 

S*  =  0  .....     (10) 
subject  to  the  conditions  : 


From  equation  (34)  of  §  121  we  find  : 

/«.*  =  T2w,         T)  +  R/w&      •         •         •     (12) 


Again,  we  put  everything  in  brackets,  except  B/  /«£,-,  equal  to//, 
where  /•  is  a  function  which  does  not  depend  on  the  composition 
of  the  mixture  : 

Thus  f«  =  ^?  ....     (18) 


m*  =  T2w,-(/f  +  R/n£)        .         .         .     (14) 
For  a  small  virtual  change  conforming  to  (11)  : 


.'.  for  equilibrium,  conditions  (10)  and  (11)  give  : 

2(f{  +  B/»6)5»,  +  2HiR8te£-  =  0       .         .     (15) 

Now  tiiblnf;  =  n,bln  •=+•  =  n,  —  -  =  bn; 

V  n, 

since  SV  =  0 

/.  the  condition  of  equilibrium  becomes  : 

2,-(ff  +  R  +  B/»^)5»,.  =  0 
or  if  we  put  : 

/f  +  R=.7i         .         .         .         .     (16) 

where  fit  like  /-,  does  not  depend   on   the  composition  of  the 
mixture,  we  have  finally  : 

2(7;  +  R/»&)8w,  =  0  .         .         .     (17) 

The    conditions    of    constant    volume  and  constant  pressure 

coincide  when  there  is  no  change  of  total  volume  during  the 

reaction  at  constant  pressure.     The  condition  8V  =  0  is,  however, 

equivalent  to  the  condition  that  there  is  no  change  in  the  total 


328  THERMODYNAMICS 

number  of  molecules.     For  if  Vt  is  the  volume  of  the  i-th  gas 
under  the  pressure  p, 

v  =  »,RT 

P 

"RT 
.-.  8V,-  =  —  8»;  (p  constant) 

.-.  if  SV;  =  0  then  5;if  =  0, 
and  if  28V,  =  8V  =  0,  then  3fin,  =  bn  =  0. 

A  compound  gas  (HC1,  HI,  NO)  produced  from  its  gaseous 
components  without  change  of  volume  is  called  a  //as  without 
condensation. 

In  this  case,  equation  (17)  can  be  derived  from  the  canonical 
potential  equation  by  putting  : 


RT 


143.     Dissociation  and  Mass  Action  in  Gases. 

In    the    applications    of    the    thermodynamic    equations    of 
equilibrium  to  gaseous  systems  we  shall  take  8n(  in  : 

S(9i  +  Rlnc^bn.;  =  0          .         ,         .     (1) 

positive  when  it  refers  to  a  substance  produced  in  the  chemical 
reaction  considered,  and  negative  when  it  refers  to  a  substance 
consumed. 

Further,  if  the  absolute  amount  of  the  change  of  any  one  com- 
ponent is  fixed,  those  of  all  the  others  are  determined  by  the 
stoichiometric  coefficients  vlt  v2,  .  .  .  in  the  equation  : 
.  .  .  ^f  r3A3  +  r4A4  -j-  .  .  . 


where  A  is  a  positive  magnitude  independent  of  the  *>'s. 

Thus       8ni  :  8«2  :....=  vl  :  v2  :.....      .         .         .     (2) 

and  (1)  can  be  written  : 

2(9,  +  R//ic,>i  =  0  .        .         .     (3) 

where  the  convention  as  to  the  sign  of  vi  is  the  same  as  that 
for  5/t;. 

If  we  compare  (3)  with  the  chemical  equation  of  the  reaction 
we  arrive  at  the  simple  rule  that  the  concentrations  in  the 
equilibrium  state  at  a  given  temperature  and  pressure  must  have 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    329 


such  values  that  whenever  the  magnitude  (9,  +  Rf'ic,)  is  sub- 
stituted for  the  chemical  symbol  A,  in  the  reaction  equation,  the 
resulting  thermodynamic  equation  is  verified. 

We  shall  call   T(9f  +  R'H<",)  the  molecular  chemical  potential 
of  the  j-th  gas  in  the  mixture,  and  denote  it  by  /Z,  : 

£,.  =  T(9(  —  R/iie,-)  =  /2°,+  RT/HC,  .  .  (4) 
where  p.'-  is  a  function  of  T  and  ^>  alone,  and  is  independent  of 
the  concentrations. 

The  equation  of  equilibrium  is  then  : 

A*i*fi  +  W'2  +  •  •  •  =  2/Zji'j  =  0  .         .     (5) 

If  we  separate  the  terms  of  (3)  we  have  : 

2^c,=  -^,9,   ....     (6) 

.Sri-pi 

or  c^ca"2  .  .  .  =  e~    R   .         .         .         .     (7) 

From  the  properties  of  9,  we  see  that  the  exponential  factor 
is  constant  at  a  fixed  temperature  and  pressure  : 

<-¥*  =  K,t        ....    (8) 
KpiT  is  called  the  equilibrium  constant,  and  the  equation  : 

Cl»c<F  .  .  .    =  KP>T  .         .         .         .     (9) 

which  states  that  the  product  of  the  concentrations  of  the  various 
constituents  of  the  equilibrium  mixture,  each  raised  to  the  power 
of  the  (positive  or  negative)  coefficient  of  the  symbol  of  that 
molecular  species  in  the  chemical  equation,  is  equal  to  a  constant 
at  a  fixed  temperature  and  pressure,  is  the  well-known  Law  of 
Chemical  Mass  Action,  deduced  on  kinetic  grounds  by  Guldberg 
and  Waage  in  1864  (Ostwald's  Klassiker,  Xo.  104).  The  deduction 
from  the  principles  of  thermodynamics  was  effected  by  A.  Horst- 
mann  (Ostwald's  Klassiker,  No.  137)  in  1873,  J.  TViilard  Gibbs 
in  1876  (loc.  cit),  J.  H.  van't  Hoff  in  1886  (Ostwald's  Klassiker, 
No.  110),  and  M.  Planck  in  1887  (H'ied.  Ann.  1887). 
Thus,  if  we  consider  the  reaction  : 


we  have,  if  the  suffixes  1,  2,  3  refer  to  H2SO4,  SO3,  H2O  :  Vl  =  —  1,  i*  =  +  1, 

"3   =   +   1, 

.-.  <*?  =  K 

Ci 

At  each  temperature  and  pressure  K  will  have  a  fixed  value,  independent 
of  ct,  ca,  fs  : 

_  "i  _  "a  _  »3 


330  THEKMODYNAMICS 

We  see  from  equation  (4)  that 


....     (10) 
where  *  is  the  total  potential  of  the  gas  mixture,  for  : 


(<I>  here  denotes  what  we  previously  called  >»</>). 

If  to  any  homogeneous  gaseous  system  we  suppose  an  infini- 
tesimal quantity  bnt  mols  of  a  specified  component  to  be  added,  the 
mass  remaining  homogeneous  and  its  temperature  and  pressure 
remaining  unchanged,  the  increase  of  the  thermodynamic  poten- 
tial of  the  mass,  divided  by  &nh  is  defined  as  the  molecular  chemical 
potential  of  that  gas  in  the  mixture  considered. 

If  the  reaction  occurs  at  constant  temperature  and  volume  we 
have  as  the  condition  of  equilibrium  : 

2(/T+  Rto£i>i  =  °  •  •         •         •     (11) 

It  can  readily  be  shown  that : 

( — )      — (  —  )       =  (       ) 
Voni/  P,T      \9»i/  V,T       \8%/  s,v 

where  *  is  the   total   free   energy  of   the   mixture,  and    U  its 
total  intrinsic  energy.     Thence  the  equations  : 


are  three  different  ways  of  defining  the  chemical  potential. 

It  is  also  easy  to  deduce,  from  the  definitions  of  c  and  £  (§  120), 
the  relation  : 

/T}T\  v 

KAT  =  KV)Tx(^)  .         .         .     (14) 

where  2?',  =  v          .         -.         .         .     (15) 

is  the  increase  of  the  number  of  rnols  in  the  reaction. 

144.     Maximum  Work  of  a  Gas  Reaction. 

If  a  chemical  reaction  occurs  spontaneously,  the  available 
energy  of  the  system  necessarily  diminishes  by  an  amount  equal 
to  the  work  which  could  be  done  by  the  system  if  the  given 
change  were  executed  reversibly.  If  the  reaction  occurs  at  con- 
stant temperature,  this  is  equal  to  the  diminution  of  free  energy 
of  the  system,  this  being  the  energy  available  at  constant  tempera- 
ture. It  is  usual  to  refer  to  the  work  available  at  constant 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS     331 

temperature  in  a  reversible  change  as  the  maximum  ivork  of  the 
change,  AT : 

AT  =  *!  -  *2  =  -  A*  .  .  .  .  (1) 
Its  value  depends  only  on  the  initial  and  final  states  and 
has  a  definite  value  for  a  given  temperature  at  which  a  rever- 
sible isothermal  process  is  executed,  and  given  terminal  states. 
Thus  if  2  mols  of  hydrogen  gas  at  a  given  volumetric  molecular 
concentration  HI,  and  1  rnol  of  oxygen  gas  at  a  given  concentra- 
tion H2,  are  converted  into  2  mols  of  steam  at  a  given  concen- 
tration Ha,  all  the  gases  having  the  same  temperature,  the 
maximum  work  has  one  definite  value,  no  matter  how  the  process 


FIG.  65. 

is  executed  provided  only  that  it  is  isothermal  and  reversible.  A 
very  instructive  way  of  carrying  out  this  imaginary  process  was 
used  by  J.  H.  van't  Hoff ;  it  depends  on  the  properties  of  semi- 
permeable  membranes. 

Let  us  take  the  example  just  considered,  and  calculate  the 
maximum  work  of  the  process.  At  the  given  temperature  there 
can  exist  an  equilibrium  state  between  the  three  gases : 

'  2H2  +  02^2H20 

which  is  definite  if  either  the  total  pressure  or  the  total  volume 
of  the  system  is  specified.  We  will  consider  the  latter  fixed  by 
enclosing  arbitrary  amounts  of  the  three  gases  in  a  rigid  box 
(Fig.  65)  at  the  given  temperature.  The  system  then  settles 
down  to  a  perfectly  definite  state  in  which  the  equilibrium  con- 
centrations of  the  three  gases  are,  say,  &,  £2,  £3- 

Now  let   us  suppose  the  sides   of  the  box  fitted  with  three 


332  THERMODYNAMICS 

cylinders,  communicating  with  the  interior  through  three  semi- 
permeable  diaphragms,  which  can  be  covered  with  impervious 
diaphragms  when  required.  We  cover  the  semipermeable  dia- 
phragms with  the  impervious  ones  and  put  into  two  cylinders 
the  gases  H2  and  02  in  the  initial  state.  These  are  now  expanded 
(or  compressed)  isothermally  and  reversibly  until  their  concen- 
trations are  equal  to  those  in  the  box. 

The  amounts  of  work  done  are  -  ZRTln  &/E  and  —  RT  In  &/E, 
respectively  (§§  79,  122). 

All  the  impervious  diaphragms  are  now  removed,  and  the  two 
gases  slowly  compressed  into  the  box  under  the  constant  pres- 
sures, the  steam  produced  being  removed  as  fast  as  it  is  formed 
through  the  third  semipermeable  diaphragm  ,  so  that  the  equilibrium 
mixture  remains  unchanged.  The  amounts  of  work  done  are 
—  2RT,  —  RT  and  +  2RT  for  the  2H2,  02,  and  2H20,  respectively. 

The  membrane  of  the  steam  cylinder  is  now  closed  by  the 
impervious  shutter  and  the  steam  expanded  (or  compressed) 
isothermally  and  reversibly  until  it  has  the  concentration  H3  ; 

the  work  done  is  —  2RT^t  ^. 
£3 

The  total  amount  of  work  done  in  the  process  is  : 


AT  =  RT   a/w        +  In        -  2to  =?    -  RT 
Ci  C2  C§/ 


-2  -  RT/w       -2  -  RT  .        .        .     (1) 

~3  €3 

If  we  keep  HI,  S2  and  H3  constant,  but  alter  the  amounts  of 
the  gases  in  the  equilibrium  box,  say  by  pumping  in  more 
hydrogen,  the  value  of  AT  cannot  alter  (since  the  equilibrium  box 
is  left  in  the  same  state  after  the  process),  and  it  depends  only 
on  the  initial  and  final  states  of  the  reacting  gases.  Hence  at  a 


given  temperature      FF.    must  be  a  constant,  i.e.,  the  composi- 

tion of  the  mixture  in  the  equilibrium  box  readjusts  itself  so  that 
this  product  maintains  a  constant  value,  independent  of  the 
absolute  amounts  of  the  substances  present. 

In  accordance  with  a  previous  convention(§  143)  we  shall  put  : 


RTln          =  RT/wK 
ti  Ca 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    333 

i.e.,  in  forming  the  sum  2r,-  we  take  the  terms  relating  to  the 
products  of  the  reaction  considered  as  positive.     Then 

AT  =  RT/nK  —  RT2»,7nE,  +  RT2v,       .         .     (2) 
Thus,  for  the  maximum  work  of  the  formation  of  hydrogen 
chloride  we  have  : 

H2  +  Cla  =  2HC1 


(AT)HC1  =  RT/»--  -  RT/n 


If  AT  is  known  (say  from  measurements  of  electromotive  force) 
we  can  calculate  K,  and  vice  versd. 

If  the  gases  are  taken  with  initial  concentrations  equal  to 
those  in  the  equilibrium  state,  the  maiimurn  work  due  to  the 
chemical  change  vanishes,  and  there  remains  only  the  part  due 
to  the  change  of  volume  : 

RTSi-i/wH,-  =  RTSi'M,  =  RT/«K 
/.  AT  =  RT2i> 

When  the  gases  are  contained  in  large  reservoirs  of  fixed 
volume,  so  large  that  the  extraction  or  addition  of  the  reacting 
amounts  does  not  appreciably  alter  the  concentrations  H,  the 
maximum  work  is 

AT  =  RTtoK  —  SV.-/HS,  .  .  .  (2«) 
since  the  work  done  in  withdrawing  v  mols  from  the  reservoir  is 
rRT,  and  that  done  in  passing  this  into  the  equilibrium  box  is 
—  rRT.  The  work  due  to  change  of  volume,  RT5>(!  therefore 
vanishes,  as  is  otherwise  obvious  since  the  total  volume  remains 
constant  (Xernst,  Theoretical  Chemistry,  Eng.  trans.,  1904,  p.  646). 
This  value  is  sometimes  called  the  maximum  work  for  isochoral 
execution  of  the  process. 

Again,  if  we  consider  the  initial  substances  in  the  state  of 
liquids  or  solids,  these  will  have  a  definite  vapour  pressure,  and 
the  free  energy  changes,  i.e.,  the  maximum  work  of  an  isothermal 
reaction  between  the  condensed  forms,  may  be  calculated  by 
supposing  the  requisite  amounts  drawn  off  in  the  form  of 
saturated  vapours,  these  expanded  or  compressed  to  the  concen- 
trations in  the  equilibrium  box,  passed  into  the  latter,  and  the 
products  then  abstracted  from  the  box,  expanded  to  the  con- 
centrations of  the  saturated  vapours,  and  finally  condensed  on  the 
solids  or  liquids.  Since  the  changes  of  volume  of  the  condensed 
phases  are  negligibly  small,  the  maximum  work  is  again  : 
AT  =  —  RTSvjteS,  +  RT/nK 


334  THERMODYNAMICS 

where  HI,  H2,  .  .  are  the  concentrations  of  the  saturated 
vapours,  and  InK  =  Si^w£,  where  £  is  the  concentration  in 
the  equilibrium  gas  mixture.  In  this  way  we  can  calculate  the 
maximum  work  of  a  condensed  reaction  when  we  know  the 
vapour  pressures  of  the  various  substances,  and  K,  for  : 
Pi  =  S,RT 

(Cf.  Nernst,  Recent  Ap2)lications  of  Thermodynamics  to  Chemistry, 
1907.) 

The  same  equations  may  be  derived  from  the  consideration  of 
the  free  energy  *.     For  by  definition  : 

'  AT=-A*  =  2*Ila  +  *(^-2*H>0 

where  ^j^  denotes    the  free  energy  of   a  mol  of  free  gaseous 
hydrogen  at  a  given  temperature  T  and  concentration  HH2  : 


Thus  AT  =  BT/n  +  T[2/H2  +  J02  -  2/H2o]         .     (3) 

~  H20 

or  generally  : 

AT  =  —  (BTS^/nE,-  +  T2v,)       .'  <*•        •     (3") 
with  the  previous  convention  as  to  2^. 

The  term  in  brackets  in  (3)  may  be  calculated  if  we  use  the 
relation  between  the  chemical  potentials  obtaining  ichen  a  mixture 
of  the  three  qases  is  in  equilibrium,  viz., 

2/iH2o  =  2/ZH2  +  ySo2 

where  /Zn2  is  the  chemical  potential  per  mol  of  gaseous 
hydrogen  in  the  mixture  at  the  temperature  T  and  equilibrium 
concentration  £H2,  i.e.  (§  143)  : 

£H2  =  TjH2  +  RT/n^H2 
where  /Ha  =  /H2  +  R 

/.  T[2/H2  +  /o2  -  2/H2o]  =  BT  -  BT/w  %^*  =  RT  +  RT/»K 

£  H20 

where  K  is  the  equilibrium  constant. 

Generally,  T2^/7  =  BTS^  +  BT/nK. 

Thus  AT  =  RT/n  "J**0*  +  BTtoK  +  RT 

*  H2O 

or  generally    AT  .=  —  RTS^»S,  +  BT/nK  -  BT2i-, 
which  are  the  equations  just  obtained. 

The  maximum  work  of  a  chemical  reaction  is  therefore  largely 
dependent  on  the  initial  concentrations  of  the  gases  ;  the  heat  of 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS     335 

reaction  at  constant  volume  is  independent  of  the  concentrations, 
since  the  intrinsic  energy  of  an  ideal  gas  is  independent  of  the 
volume. 

It  may  be  remarked  that  a  certain  amount  of  confusion  exists  in  the 
literature  respecting  the  magnitude  we  have  called  the  maximum  work. 
Thus  Nernst  and  Haber  always  omit  the  term  ET2^,-  which  belongs  to  the 
work  done  on  the  system  by  the  change  of  volume.  Although  the  latter 
author  (Thermodynamics  of  Technical  Gas  Reactions,  Eng.  trans.,  p.  54) 
observes  that  this  term  should  be  included,  he  decides  to  omit  it  from  his 
equations  on  the  ground  of  "  simplicity."  It  is  not,  however,  a  matter  open 
for  definition  in  this  way,  since  the  maximum  work  of  a  change  must  always 
be  calculated  for  a  definite  choice  of  conditions.  Thus,  the  work  done  by 
the  galvanic  cell  consuming  hydrogen  and  oxygen  gases  at  its  electrodes, 
and  producing  liquid  water,  will  include  the  external  work  done  by  the 
atmosphere  in  compressing  the  gases  2H2  and  02,  and  this  term  must  be 
included  if  we  wish  to  calculate  the  electromotive  force  of  the  cell  (cf.  Haber, 
loc.  cit.  ;  also  later  §  205).  Nernst's  equation  (2«)  is  deduced  for  the  gases 
taken  from  large  reservoirs  (i.e.,  at  constant  volume),  and  is  therefore  not 
general,  since  no  actual  gas  reactions  with  a  change  in  the  total  number  of 
molecules,  and  none  whatever  when  the  products  are  liquids  or  solids,  or 
remain  dissolved,  occur  in  this  way.  The  "  simplicity  "  is  far  outweighed 
by  the  uncertainty  as  to  the  actual  conditions  implied. 


145.     Influence  of    Temperature  and  Pressure  on    the  Equi- 
librium in  a  Gaseous  System. 

The  equations  of  equilibrium  in  a  gaseous  system  are  : 

2v(/»e,  =  —  =^  =  /nKp>T  Q>,T  const.)     .         .     (1) 

2^n6  =  -  ?^  =  /wKVtT  (V,T  const.)    .         .     (2) 

If,  for  brevity,  we  omit  the  suffix  T  which  is  common  to  both 
K's  we  find,  by  partial  differentiation  : 

dlnK\  I   3  _ 


(5) 


336  THEEMODYNAM1CS 

where  as  before  : 


«,,<T) 


=  Ul*  -  TSj,"  +  f  CJMT  -  T  p|S. 


Thence  we  readily  find  : 

=  -  (6) 

T  P 

/  agA             Ui«  +  CKWT  +  RT           U(/)  ,  RT  W(, 

V8T/,>= J    T2  ~  = ^ =—-^5-.    (7) 

since   U^  +  JCi/'dT  =  U(0   by   Kirchhoff's  theorem  (§  58), 
and  U(ii  +  pf  =  U("  +  RT  =  W("  by  definition  (§  25) 


-    (8) 


We  now  substitute  the  partial  differential  coefficients  in  (3), 
(4),  (5)  : 

ai|<       =  -=_? 
T  P  P  P 

dlnK\  1  /      W 


since  2i>jWw  =  Qp  =  heat  of  reaction  at  constant  pressure, 
and  2^-U1''  =  Q.F  ^  heat  of  reaction  at  constant  volume. 
The  influence  of  a  simultaneous  change  of  temperature  and 

pressure  is  now  calculable  : 


f2f?T    .         .        .     (12) 
146.     Influence   of  Pressure  on  Gaseous  Equilibrium. 


The   influence   of  pressure   increases   numerically   with    the 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    337 

contraction,     and     decreases     numerically     with     the     initial 
pressure. 

If  V,  V  are  the  total  volumes  of  the  initial  and  final  systems, 
2i>  and  2i/  the  total  numbers  of  mols  in  these  respectively  : 
pV  =  2i»  .  RT 
p\'  =  2»f  .  RT 
.-.  p(\'  -  V)  =  (2«/  - 


RT        ~  RT 


We  can  also  write  (1)  in  the  form  : 


If  we  put  Inp  =  x,  /wKi  =  i/,  the  curve  /nK  =f(lnj))  will  be  a 
straight  line  having  a  positive,  zero,  or  negative  gradient  according 

as  v  =  0. 

(1)  There  is  expansion,  V  >  V ;  then  J/wK  <  0,  i.e.,  increase  of 
pressure  favours  the  production  of  system  V  with  the  smaller 
volume. 

(2)  There  is  contraction,  \'  <  V ;  then  rf/»K  >  0,  i.e.,  increase  of 
pressure  favours  the  production  of  system  V  with  the  smaller 
volume. 

(3)  There  is  wo  change  of  volume,  V  =  V,  then  r//wK  =  0,  i.e., 
change  of  pressure  has  no  influence  on  the  equilibrium. 

If  p  is  in  atm.,  V  in  litres,  then  R  =  0'08207  1.  atm. 
The  equation  representing  the  influence  of  pressure  on  gaseous 
equilibrium  is  due  to  Planck. 

147.     Influence  of  Temperature  on  Gaseous  Equilibrium. 

The  integration  of  the  two  equations : 

=  ^^2  (Reaction  Isochore)  ....     (1) 

=  TT^  (Reaction  Isopiestic) ....     (2) 

leads  to  some  of  the  most  important  applications  of   thermo- 
dynamics to  chemistry.     We  take  them  in  order. 


338  THERMODYNAMICS 

(1)  The  Reaction  Isochorc. 

\^r)  v  =  RT^ 
where          /«KV  =  *'i/»£i  +  r2/»i£2  +  .  .  =  £"//»£        .         .     (3) 

Thence  foKy  =  g  |  jj|j  dT     .        .        .         .     (4) 

The  integration  is  possible  when  Qt.  is  known  as  a  function  of 
temperature.     But,  according  to  Kirchhoff's  theorem  : 

Q^Q-'+^T/-^)^!       .         .         .     (5) 

V  —  r,.WT 


- 

"  RT         ~      RT2 
(a)  If  IV  =  T          .          .        -.     .    .         (7) 

then  //»K,  =  -  1^  +  const.        -.         .        .     (8) 

where  Qr  is  the  heat  of  reaction  at  all  temperatures. 

Assumption  (7)  implies  that  the  molecular  heat  of  each  com- 
pound gas  is  the  sum  of  the  atomic  heats  of  its  constituents  (a 
hypothesis  introduced  on  the  basis  of  experiment  by  Delaroche 
and  Berard  (1813),  and  afterwards  defended  by  Buff  (1860),  and 
Clausius  (1861)  ),  and  also  that  the  molecular  heats  are  indepen- 
dent of  temperature.  This  rule  is  only  very  approximate,  the 
deviations  sometimes  exceeding  30  per  cent.,  although  (8)  is  often 
useful  over  a  small  range  of  temperature. 

If  we  put  "  const."  —  a,  and  Q,./R  =  const.  =  1>, 

__b_ 

Kv  =  ae  T  .     :    .         .         .       (9) 

For  two  temperatures  Tlf  T2  (T2  >  TI)  : 


.     '..    (10) 

(/>)  if  r/  -  rv  =  const.      .       .      .    (ii) 

i.e.,  the  specific  heats  are  independent  of  temperature  (hypothesis 
of  Clausius),  then  : 

/»K(.  =  -  §!'  +  r<-'  ~  F(  ;/,T  +  const.         .         .     (12) 
or  Iv=a<fTTc        .         .         .         .     (13) 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    339 

(c)  If  the  specific  heats  are  linear  functions  of  temperature  : 

-rr  =  Si>Xa,  +  2ftT)       ....     (14) 
then       Qr  =  Qj,r'  +  (Sy/o/  —  2r,a,)T  +  (2r//8/  —  2^)1* 

=  Q;r)  +  aT  +  £T2         .         .         .     (15) 

and  /«Kr  =-^+|//<T+  |T  +  const.    .         .     (16) 

where  a  =  S^-'a/  —  2f,a,  ;  /8  =  2i>/&'  —  2r,-/8,-. 

(2)  Reaction  Isopiestic. 


f-dT        .        .         .     (17) 

Now  Qp  =  Qr  +  rRT     ....     (18) 

".*.  Qp  and  Qr  are  equal  in  two  cases: 

(i.)  When  v  =  0,  i>.,  the  reaction  proceeds  without  change  of 
volume, 
(ii.)  When  T  =  0,  i.e.,  at  the  absolute  zero, 


Hence  Qp  =  Q;r'  +  I  (F/  -  rp)dT 

.-.  Qp  =  Qirt  +  f  (IV  -  rr)dT  +  t-KT     . 

since  Cp  =  Cr-+R 

and  Fr  =  S^-C*  ;  rp  =  2^'. 

(o)  if  r;  -  rp  =  o 

/»Kp  =  -^  +  const.  .         .         .         .     (21) 

Now  Tp'  —  Tp  can  be  zero  only  if  we  have  simultaneously 

TV  -  Tr  =  0 

and  v  =  0 

in  which  case,  Kp  =  Kr 

If,  however,  F/  -  Fr  =  0,  but  v  >  0 

then  lnKp  =  -  ^  +  rZwT  +  const.         .         .     (22) 

(6)  If  Tp'  -  Tp  =  const. 

(28; 
z  -2 


340  THERMODYNAMICS 

If  r/  —  F^  =  0,  this  passes  over  into  (22). 

(c)  If  re  =  Si;,  (a,  +  2&T) 


T+  const.     .         .     (24) 


where  the  symbols  have  the  usual  significance. 

If  v  =  0,  this  coincides  with  the  isochore  equation  (16). 

If  we  consider  both  Kr  and  K,,  together,  and  drop  the  suffixes, 
we  see  that  : 

if  Q  >  0,  dlnK  >  0,  or  K  increases  with  T, 
if  Q  <  0,  dlnK  <  0,  or  K  decreases  with  T, 
if  Q  =  0,  dlnK  =  0,  or  K  is  independent  of  T. 

Thus  K  changes  with  rise  of  temperature  in  such  a  icay  as  to 
favour  the  reaction  which  proceeds  with  absolution  of  heat.  (Yan't 
Hoff's  Law  of  Mobile  Equilibrium,  1886.) 

To  avoid  the  confusion  which  sometimes  arises  we  may  remark 
that  the  sign  of  Q  is  fixed  in  accordance  with  the  convention  as 
to  which  substances  are  taken  as  forming  the  initial  system. 
Thus,  if  we  consider  the  reactions  : 


a)  2H2  +  02  =  2H20,  then  Ka  =   2     H*>     ,  and  Qa  <  0  ; 

' 


/3)  2H20  =  2H2  +  02,  then  K^  =  *'  and  Q«  >  0. 

C  H20 

In  both  cases  rise  of  temperature  (at  constant  volume  or  con- 
stant pressure)  favours  the  dissociation  of  steam,  since  this 
reaction  tends  to  cool  the  system  down  to  the  initial  temperature. 

148.     Gibbs's  Dissociation  Equation. 

For  a  simultaneous  change  of  temperature  and  pressure  : 
dlnK>} 

" 


/.  InKp  =  —  vlnp  -f-  p 

The  integration  is  possible  when  Q/;  is  known  as  a  function  of 
temperature,  i.e.,  when  Qr  for  one  temperature  and  the  specific 
heats  at  all  temperatures  in  the  range  considered,  are  known. 
We  see  at  once  that  the  desired  result  is  obtained  simply  by 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    341 

adding  —  vlnp  to  the  equations  (17)  —  (24)  of  the   preceding 
section. 

If  the  specific  heats  are  independent  of  temperature  : 

/nK,  =  -  g|r+  T'  ~  R  +  rR  toT  -  vlnp  +  const.         .     (8) 
Put  F/  —  Fr  =  const.  =  b 

then  Kp.T  =  a.T*+r«~SfP~l  •  •  '  •  <4) 

where  Ina  is  the  "  const."  of  (3). 

Equation  (4)  is  the  Dissociation  Equation  of  Willard  Gibbs 
(1876). 

If  the  specific  heats  are  linear  functions  of  temperature  : 


/»K,  =  -     jL  +  "  f  »T  +      T  -  rlnp  +  const. 

0_+      _4t+p_T 

orKp,T  =  A.TR       e    *T     R    p~v        .         .     (5) 

The  extension  to  include  higher  powers  of  T  is  obvious. 

We  observe  that  all  constants  in  (4)  or  (5)  can  be  determined 
by  measurements  of  the  thermal  properties  of  the  system,  with 
the  exception  of  a  or  A,  which  are  indeterminate  from  the  point 
of  view  of  classical  thermodynamics. 


149.     Applications  of  the  Equations. 

The  extent  of  dissociation  of  a  gas  is  usually  determined  from 
the  density.     Let 

D  =  density  when  no  dissociation  occurs, 

A  =  observed  density  of  partially  dissociated  gas, 

d  =  density  of  completely  dissociated  gas, 
all  reduced  to  normal  temperature  and  pressure. 

Let  each   molecule   of  the   original  gas  break  down   into  x 
molecules  on  dissociation,  then 

*  =  i  •        •        W 

We  have : 

d<A<D 
unless  x  =  1. 

If  the  fraction  of  the  original  gas  which  ^is  dissociated,  or  the 


342  THERMODYNAMICS 

extent  of  dissociation,  is  denoted  by  7,  we  have,  from  A.  mols  of 
gas  originally  taken 

(1  —  7)Amols  undissociated 

7-rA  mols  dissociated 

/.     total  number  of  mols  after  dissociation  =  [1  -J-  (•*-'  — 
-D  f  No.  of  mols  after  dissociation 


No.  of  mols  before  dissociation 
volume  after  dissociation 
volume  before  dissociation 
_     density  before  dissociation 

density  after  dissociation 
(a;-l)7]  _  D 
A  ~  A 


If  A  =  D/x,  then  7  =  1,  i.e.,  the  dissociation  is  complete  ;  if 
A  =  D,  then  7  =  0,  i.e.,  there  is  no  dissociation  (if  x  =£  !)• 

The  various  types  of  dissociation  may  be  classified  according 
to  the  value  of  x. 

(1)  x  =  1.  In  this  case  the  extent  of  dissociation  cannot  be 
determined  from  the  density,  since  the  latter  is  always  equal  to 
that  of  the  undissociated  gas  ;  it  must  be  estimated  in  some  other 
way  (e.g.,  by  chemical  analysis  if  the  system  can  be  cooled 
sufficiently  rapidly  to  avoid  alteration  of  the  composition  in  the 
equilibrium  state). 

The  equilibrium  equation  is  : 

^3  K  TT-  T^ 

—3-  ==  K,  =  K,  =  K 

where  the  suffixes  1,  2,  3  refer  to  the  dissociating  gas  and  the 
products,  respectively,  e.g., 

12HI  =  H2  +  I2. 

Also  &  =  pcVBT,  £2  =  2>c2/RT,  £,  = 
where  p  is  the  total  pressure, 


The  partial  pressures  are  : 

Pi  —  pci,  P*  =  PC*  Pa  =  pea 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS     343 

Corollary  1. — The  total  pressure  has  no  influence  on  the 
equilibrium. 

Corollary  2. — The  addition  of  aninerb  gas  has  no  influence  on 
the  equilibrium,  since  £1}  £2,  £3  are  all  altered  thereby  in  the 
same  ratio. 

The  total  volume  remains  constant,  hence  the  partial  pressures 
are  proportional  to  the  numbers  of  mols  of  each  gas.  Thus,  of 
2A  mols  HI  there  will  be  2yA  mols  dissociated,  and  there  are 
formed  y\  mols  H2  and  7  A  mols  I2,  whilst  2(1  —  7)A  mols  HI 
remain.  The  total  number  of  mols  is  constantly  2A,  hence  : 


If  pure  HI  was  initially  present : 

Pz  =  PS  =  p'  say, 

~  1  +  2\/K  '  lh  '     1  +  2A/K 

If,  however,  A2  mols  H2  and  A3  mols  I2  where  A2  =J=  A3,  are  intro- 
duced into  a  closed  space,  and  heated  at  a  constant  temperature 
till  the  system  is  in  equilibrium  with  2Aj  mols  HI  produced, 
we  have : 

A2  mols  H2  A3  mols  I2  no  HI       at  the  start 

A2  —  AX  mols  H2        A3  —  A:  mols  I2        2A  mols  HI  at  the  finish 
/.  if  pi,  pi,  2}s  are  the  respective  partial  pressures, 

A2  —  \i  A3  —  AI  2Ai 

p%  —  r —        p  ',  2*3  — j 2*  5  PI  —  , P' 

A2  +  A3  A2  -f-  A3  A2  -(-  A3 

A2  —  AX  is  determined  by  measuring  the  volume  after  absorption 
in  water ; 

A3  —  A!  and  2A  are  determined  by  titration.     Then 
(A2  —  A!)  (A3  —  Ax) 

4A2  -  *2 


A2  +  A3  /    (A2 

~  21  -  4K         V   41  - 


A2A3 

2(1  -  4K2)      V   4(1  -  4K2)2  ~  1  -  4K2 
(2)  x  =  2  (binary  dissociation)  : 

AB  ^  A  +  B 
123 


344  THERMODYNAMICS 


pi 

where  p  =  total  pressure  •=  pi  +  jp2  +  l>s- 

If  there  is  no  excess  of  the  products  of  dissociation  : 

c2  =  c3  ;  and  p2  =  pa  =  /  say,  pi.  =  p  —  2/ 


In  the  case  of  nitrogen  tetroxide  : 

NA^N02  +  NOa 

fche  two  products  are  identical,  and  if  p'  is  the  total  partial  pres- 
sure of  N02,  p'  =  2pz  =  2j?3 

•'•    (P  —  Pi)2  =  4  K2fti 


p 

, 

27  A 


Also  PL  =  1~fy,  where  7  =  ^  A 

' 


(D  -  _ 

"    (2A  -  D)D  ~ 


The  equilibrium  is  largely  influenced  by  the  total  pressure. 
By  solving  for  A  we  find  :  _ 


K 
where  K'  =  K2D. 

Gibbs's  equation  may  be  applied  in  the  form  : 

loci  Ki  =  -  Q^-  B/wT  —  vlnp  +  A 

where  B  =  —  -Ll—",  and  A  is  an  arbitrary  constant. 

(D-A/     _  (M  -  A)2 
«i  -•*•/!>«    (2A-D)D    ~4(A-dX 

where  f/  =  density  of  N02  (etc.) 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS     345 


According  to  Gibbs  the  term  with  InT  may  be  omitted  if  it  is 
compensated  in  the  values  of  A  and  Q0,  regarded  as  arbitrary,  thus, 
with  ordinary  logarithms : 

(D  —  A)2    ._  ,     (2d  —  A)2  _       M 


%(2A-D)D 
where  M  and  N  are  constants. 

0.  Brill  (Zeitschr.  physik.  Chem.  5', 
equation : 

loci 


721,  1907)  has  used  the 


T  - 


Gibbs  also  showed,  on  the  basis  of  some  experiments  of  Play- 
fair  and  Wanklyn,  that  the  equation  applies  when  p  is  reduced, 
not  by  decreasing  the  mechanical  pressure  on  the  system,  but  by 
admixture  with  an  inert  gas.  (Cf.  Dixon  and  Peterkin,  Trans. 
Chem.  Soc.,  1899,  p.  613.) 

(3)  If  x  =  f  we  have  as  examples  the  very  important 
equilibria  : 

2CO-2  ±z  2CO  +  02 
2H20  ^  2H2  +  02 

a  description  of  the  experimental  study  of  which  will  be  found  in 
Haber's  Thermodynamics  oj  Technical  Gas  Reactions. 
The  following  tables  of  the  extent  of  dissociation 

g  =  lOOy 
contain  the  results  of  the  most  recent  investigations. 

(1)  Halogen  Hydracids. 


r        T' 

HC1 

HBr 

HI 

290 

2-51  X  10  ~15 

414  X  10~s 

6"2 

500 

1-92  X  10- 

2-91  X  10-* 

15-5 

700 

1-12  X  10- 

9-93  X  10  -:i 

22-2 

900 

3-98  X  10- 

7-18  X  10~2 

27'0 

1,000 

1-34  X  10- 

0-144 

29-0 

1,500 

6-10  X  10- 

1-19 

— 

2,000 

0-41 

3-40 

— 

2,500 

1-30 

— 

— 

(Vogel  von  Falckenstein,  Zeitschr.  physik.  Chem.,  68,  3,  270.) 


346 


-THERMODYNAMICS 


(2)  Nitric  Oxide. 
2NO  —  Na  +  Oa 


T° 

Per  cent.  N2 

Per  cent.  0-2 

Per  cent.  NO 

1,811 

78-92 

20-72 

0-37 

1,877 

78-89 

20-69 

0-42 

2,033 

78-78 

20-58 

0-64 

2,195 

78-61 

20-42 

0-97 

2,580 

78-08 

19-88 

2-05 

2,675 

77-98 

19-78 

2-23 

3,200 

76-6 

18'4 

5-0 

(Nernst,  Gottinger  Nachr.,  1904,  p.  261 ;  with  Jellinek  and 
Finckh,  Zeitschr.  anorg.  Chem.,  46,  116,  1905;  49,  212,  229, 
1906.) 

(3)  Steam. 
2H20  =±  2H2  +  02 


T 

p  =  10  atrn. 

1  atm. 

o- 

1  atm. 

0-01  atm. 

1,000 

1-39  X  10  -5 

3-00  X  10  -5 

6-46 

x  io-5 

1-39  X  10-  ! 

1,500 

1-03  X  10  ~2 

2-21  X  10-2 

4-76 

x  io-2 

0-103 

2,000 

0-273 

0-588 

1-26 

2-70 

2,500 

1-98 

3-98 

8-16 

16-6 

(4)  Carbon  Dioxide. 


T 

p  =  10  atm. 

1  atm. 

0-1  atm. 

0-01  atm. 

1,000 

7'31  X  10~6 

1-58  X  10"5 

3-40  X  IO-5 

7-31  X  10~5 

1,500 

1-88  X  IO-2 

4-06  X  10  -2 

8-72  X  IO-2 

0-188 

2,000 

0-818 

1-77 

3-73 

7-88 

2,500 

7-08 

15-8 

30-7 

53-0 

(Cf.  Nernst  and  von  Wartenberg,  Zeitsclir.  physik.  Chem.,  56, 
513,  534, 548, 1906  ;  Haber,  Thermodynamics  of  Techn.  Gas  React., 
Eng.  trans.,  1  Appendix  to  Sect.  V.) 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    347 

From  these  numbers,  a  large  number  of  calculations  of 
technical  interest  can  be  made.  Further,  if  we  divide  the 
equilibrium  constant  of  carbon  dioxide  by  that  of  steam  we  obtain 
the  equilibrium  constant  of  the  water-gas  equilibrium : 

[CO]2  X  [QJ  [H20]2          Kco,  =  ( [CO]  X  [H20]]2 

[C02]2        A  [H2]2  x  [O2]      KH*°       (  [C02]  X  [H2]  / 
.    K  =  [C01XJEW)]  _     / 
[C02]  x  [Hi]       V 
Again,  if  we  divide  the  square  of  the  equilibrium  constant  for 
hydrogen  chloride  by  that  for  steam  we  obtain  the  equilibrium 
for  the  Deacon  process  of  chlorine  manufacture  : 

9        4HC1  +  02  —  2H20  +  2Cla 

Ki2  _  [H2]2  x  [C12]2      _JSiO]a_  _  [H20]2  x  [C12]2      R 
Ka  ~        [HC1]4        *  [Ha]2  X  [O2]  ~"  [HC1]4  X  [02]  ~ 
Example. — The  dissociation  of  steam  : 


-  _    p      2  —27  .  £         p         27     .  f         P 
~  RT  '  2  +  7  '  RT  *  2  +  7  '  RT 

where  7  =  extent  of  dissociation,  p  =  total  pressure 


Thermal  Data  : 

Cc  for  H20  :  5'61  +  0'000717T  +  3'12  X  10 
Ce  for  02  and  H2  :  4'68  +  0-00026T 

Qr  (at  T  =  373°)  =  115300  cal. 

/.     from  Kirchhoff's  equation  : 


Qr  -=  114400  +  2-74T  —  0'00063T2  -  6'24  X  1Q-7T3 
...  log  K  =  -  ^°*9  +  2-38  log  ^  -  1-38  x  10  -  4  (T  -  1000) 

-  0-685  X  10  -7  (T2  -  10002)  +  const. 
by  integration  of  the  reaction  isochore  (§  147). 
For  T  =  1000,  7  =  8'02  X  10-7(obs.) 

/.      const.  =  11-46 


-  1-38  X  10-4(T-1COO)  -0-685  X  10  ~7  (T2  -  1CC02). 


348  THERMODYNAMICS 

From   this  equation  the  values  in  the  table  were  calculated 
(Nernst  and  Wartenberg,  loc.  cit.}. 


150.     Maximum  Work  at  Different  Temperatures. 

In  §  144  we  have  deduced  the  expression  : 

AT  =  RTZwK  —  RT2/^wSj  ,—  RT2i-,;        .         .     (1) 
for  the  maximum  work  of  an  isothermal  gas  reaction  at  T°. 

If  the  temperature  at  which  the  process  is  executed  is  changed, 
whilst  the  amplitude  of  the  process  (§  58),  i.e.,  the  initial  and 
final  concentrations  H,  remain  unchanged,  we  shall  have: 

AT  +  Qr  =  T^T    4  •-•;•    .      .   (2) 

where  :  AT  =  maximum  work  at  the  temperature  T, 

Qr  =  heat  of  reaction  at  temperature  T  and  constant 
volume, 

-grjr  =  rate    of    increase    of     maximum   work    with    the 

temperature  at  constant  amplitude. 
Differentiate  (1)  with  respect  to  T  and  substitute  in  (2)  : 

T  =  RtaK  +  RT  **      -  RSi;,toS,  -  RSr,     (S,  const.) 


•'*  7nK  =  5f?T  +  (const')  '       '       '  (8o) 

If  we  carry  out  the  integration,  and  substitute  in  (1)  we  have 
an  equation  giving  the  influence  of  temperature  on  the  maximum 
work. 

From  Kirchhoff's  equation  : 


Thus: 

AT  =  T[^|  dT  -  RTSi^wE,  -  RT2V,  +  (const.)T     .     (4) 
If  we  put 

r;  -  r,  =  «  +  2/8T  (cf  .  §  144) 


.-.     AT  =  -  Qo  +  aT/wT  +  /9T2  -  RTSr^wS,  +  (const.)T     (5) 
where  RT2^  has  been  included  in  the  constant. 

Partial  pressures   may  be   substituted  for  concentrations  by 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS     349 

making  use  of  the  relations  of  §  122.   (Cf.  Haber,  Thermodynamics 
of  Technical  Gas  Reactions,  Eng.  trans.,  p.  61.) 

151.     External  Work  in  Dissociation  of  Gases. 

If  *,  *'  are  the  free  energies  of  a  system  before  and  after 
dissociation  at  a  constant  temperature,  the  maximum  external 
work  obtainable  is  *  —  *f.  This  may  be  calculated  directly. 
Let  us  take  the  case  of  nitrogen  peroxide  : 


Let  n  =  number  of  mols  N204  originally  taken,  7  =  degree  of 
dissociation. 

Then  if  v  =  total  volume,  we  have  : 


The  pressures  before  and  after  dissociation  are  proportional  to 
the  numbers  of  mols  (Avagodro's  theorem)  : 

pip'  =  n/[(l  -  7)11  +  27*1]  =!/(!  +  7) 

/.!/=  XI  +  7) 
Also  SAT  =  p'dv  =  p(l  +  y)dv 

But  from  Boyle's  law  p  =  BT/c  =  BTKp(  \  ~  7)       .        .     (I) 
To  find  dv  we  differentiate  (a)  : 


/.  AT 


r       p 

=    pdv  +     ypdv 

J    VI  J     VI 


=*-*•• 


152.     The  Specific  Heat  of  a  Dissociating  Gas. 

If  the  temperature  of  a  mixture  of  a  gas  and  its  products  of 
dissociation  is  raised  by  ST  at  constant  pressure  p  the  quantity 
of  heat  absorbed  Cy,5T  is  made  up  of  four  parts  : 


850  THERMODYNAMICS 

(i.)  the  heat  required  to  warm  the  undissociated  part  through 
ST  at  constant  volume,  (1  —  y)Cr  ; 

(ii.)  the  heat  required  to  warm  the  dissociated  part  through 
8T  at  constant  volume,  viz., 

(wi7C<?  +  "27CT  +  .  .  .  )ST 

where  n1}  H2,  .  .  .  mols  of  the  products  result  from  the  dissocia- 
tion of  one  initial  mol  ; 

(iii.)  the  heat  absorbed   in    performing  external  work,    i.e., 

p  (  om)  8T,  where  V  =  total  volume  ; 

(iv.)  the  heat  absorbed  in  the  further  dissociation,  Q,, 

where  Qp  =  heat  of  dissociation  (approximately  constant). 
Thus  the  observed  molecular  heat  is  given  by  : 

C,  =  (1  -  7)Cr  +  7(»iC<"  +  »&?  +  .  .)  +  P  (H)  f  +  QP 

•     P   -         •     (1) 
In  the  case  of  binary  dissociation,  e.g., 

N204  -^  2N02 

C,  =  (1  -7)Cr 
But  (§  152) 


and  (§  149  (2)  )  :    In  =  ~        +  const. 


2T2 
If  we  substitute  (2)  and  (3)  in  (1)  we  find 

p  =  (l-7)C(.  +  27CJr  +  R1  (l 
L 


If  C,,  is  observed  experimentally,  and  if  Qp  is  found  from 
measurements  of  dissociation  pressures,  then  we  can  find  7  at 
any  temperature  by  measurement  of  the  specific  heat  if  Cr,  CV, 
are  known  ;  and  conversely  if  Cc,  or  C'p1',  and  7  are  known  we 


can  find 


CHEMICAL  EQOLIBBIFM  IX  GASEOUS  SYSTEMS    351 

Measurements  in  this  field  have  been  made  by  Berthelot  and  Ogier  with 
nitrogen  tetroxide  (Aim.  de  Chim.  et  Phys.,  [v.],  30,  382  (1883)),  and  with 
acetic  acid  {ibvi.,  400),  and  some  calculations  with  reference  to  steam  have 
been  made  by  Xerost  (Verhandl.  Bcubth.  Phy*.  Ges.,  1-5,  313)  and  Levy 
{ibid.,  330),  who  utilised  the  vapour-pressure  measurements  of  Holborn 
and  Henning  (Ann.  der  Physik,  (1906),  21  ;  (1907),  Sg,  23}.  Wiedemann 
had  previously  observed  that  the  specific  heats  of  ethylbromide,  ethyl- 
acetate,  and  benzene  increase  with  temperature  at  about  the  same  rate  as 
that  of  nitrogen  tetroxide  at  20»'.  In  the  case  of  steam  it  was  assumed  that  : 

(L)  the  polymerisation  is  to  double  molecules 

(iL)  the  molecular  heat  of  the  double  molecule  is  twice  that  of  the  simple 
molecule. 

CL  also  Duheni.  Traite  </«*  J/«T<IMI</M<>  ehimique,  IL,  312  ft 
seq.,  in  which  various  possible  cases  are  discussed,  and  Swart 
(Zeit.  phfizik,  Chem.,  7,  120*  1891> 


153.     The  Shape  of  Dissociation  Curves. 

To  get  an  idea  of  the  general  trend  of  dissociation  in  a  gas. 
we  shall  consider  the  isopiestics  which  represent,  at  various 
constant  pressures,  the  density  A  as  a  function  of  temperature. 

We  may  take  the  case  of  binary  dissociation  (JT  =  *2»  as 
exemplified  by  the  reactions: 


for  which  Gibbs's  equation  takes  the  form  : 
.    (-2A  -  D)D      M  . 
ln    (D-Af  =T+faP--N   '         '         '     (1) 

We  take  the  reaction  for  which  M  is  positive,  i.e..  in  which 
heat  is  absorbed. 

As  T  increases  from  0  to  +  x,  P  meanwhile  remaining  constant. 
the  second  member  of  equation  (1  )  decreases  from  a  value  -f-  x 
to  a  finite  limiting  value  t/nP  —  X),  which  is  all  the  greater  the 
larger  is  the  value  of  P. 

If  the  temperature  is  kept  constant,  and  P  varies,  the  second 
member  of  (1)  is  all  the  greater  the  larger  is  the  pressure, 

From  a  consideration  of  the  first  member  of  (1)  we  see  that,  as 
A  increases  from  £D  (i.e.,  d)  to  D,  the  expression  increases  from 
—  x  to  +  x  (ct  H.  M.,  §  45> 

If  we  combine  the  results  of  the  investigation  of  the  first 
member  of  (1)  with  the  results  of  the  investigation  of  the  second 
member,  we  are  led  at  once  to  the  following  theorems  : 

(1)  If  a  dissociable  gas  is  heated,  at  constant  pressure,  from 


352 


THERMODYNAMICS 


the  absolute  zero  to  a  temperature  greater  than  any  assignable 
temperature  (T  =  oo  ),  its  density,  A,  decreases  continuously  from 

the  value  D  to  a  limiting  value  which  is  greater  than  the  value 

corresponding  with  complete  dissociation. 

(2)  This  limiting  density  is  all  the  less  the  smaller  is  the  value  of 
the  pressure,  and  tends  to  the  limiting  value  D/2  as  P  tends  to  zero. 

(3)  At  a  given  temperature   the  density  is   all   the  less   the 
smaller  is  the  pressure. 

From  (1)  by  differentiation  (cf.  H.  M.,  §  47) 


(2) 


also 


-  (2A  -  D)  = 


(D  -  A)2 
As  T  tends  towards  zero,  the  numerator  of  (2)  is  infinite-  of  the 


5 

t; 

2 
L 

"^'^^^x. 

\ 

•v 

X 

\ 

'  

- 

*0            160           200          240           280          320           3t 
Temp. 
FIG.  66. 

order  1/T2,  whilst  the  denominator  is  infinite  of  the  order 
hence,  by  a  well-known  theorem, 

Lim  3  A       A 


Similarly,  tor  T  -.0  :  f£  =^=  .  . '.  =  f£  =  .  .  =  0.     (4) 
so  that  the  curves  have  contact  of  infinite  order  with  the  line 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    353 

Similarly,  for  T  — >  oo ,  it  can  be  shown  that  the  curves  have 
contact  of  infinite  order  with  their  asymptotes,  i.e.,  the  horizontals 
through  the  limiting  densities. 

Each  curve  therefore  consists  of  three  parts  ;  an  initial  and  a 
final  portion  which  are  nearly  horizontal  for  a  finite  part  of  their 
lengths,  and  an  intermediate  portion  which  slopes  down  com- 
paratively rapidly  from  left  to  right.  This  means  that  the 
dissociation  with  rise  of  temperature  is  slow  at  first,  then 
increases  very  rapidly,  and  then  becomes  increasingly  slower  as 
it  approaches  asymptotically  to  the  limiting  value  for  T  =  x  . 
The  general  form  of  curve  so  predicted  corresponds  exactly  with 
the  experimental  curves,  as  will  be  seen  from  Fig.  66,  which  was 
drawn  by  Horstmann  from  the  results  of  Wiirtz  with  amylene 
hydrobromide  : 

C5H10 .  HBr  =  C5H10  +  HBr. 

154.     Experimental  Study  of  Gaseous  Equilibria. 

In  the  application  of  the  equations  deduced  in  the  preceding 
sections,  it  is  necessary  to  know : 

(1)  The  heat  of  reaction  Qc. 

(2)  The  specific  heats  of  the  various  substances,  to  calculate 
the  value  of  (T1  —  T)  required  in  the  application  of  Kirchhoff's 

equation,  and  the  evaluation  of  the  integral  for  InK : 

/• 

QT  =  Q,  +  I  <r  -  ry/T 


These  may  be  called  the  calorimetric  measurements. 

(3)  The  temperature  T  of  the  system  in  equilibrium.  This 
is  measured  in  the  usual  way  when  T  is  not  large.  The 
measurement  of  high  temperatures  has  attracted  a  considerable 
amount  of  attention  on  account  of  its  great  importance.  A 
detailed  description  of  the  methods  used  in  the  measurement  of 
high  temperatures  will  be  found  in  Le  Chatelier  and  Bouduard's 
High  Temperature  Measurements,  trans.  Burgess  (Wiley,  New 
York).  The  chief  methods  are : 

(a)  Platinum  resistance  thermometers — for  temperatures  between 
-  180°  and  1,500°  C. 

T.  A    A 


354  THERMODYNAMICS 

(b)  Thermocouples — of  copper  and  constantan  at  moderate  tem- 
peratures, or  platinum  with  an  alloy  of  platinum  and  10  per  cent, 
of  indium  or  rhodium  at  high  temperatures  (cf.  Pfund,  Phys. 
Rev.  1912). 

Very  accurate  instruments  of  types  (a)  and  (b)  for  both  scientific 
and  technical  purposes  are  made  and  sold  by  the  Scientific 
Instrument  Company,  Cambridge,  England. 

(c)  Optical  Pyrometers,  the  action  of  which  is  based  on  the 
thermodynamic  laws  of  radiation  (cf.  Le  Chatelier  and  Bouduard, 
loc.  cit. ;  Waidner  and  Burgess,  Optical   Pyrometry,  Report  of 
Bureau  of  Standards,  Washington  ;  Planck,  Tkeorie  tier  Warmes- 
trahlnny,  Leipzig,  1906  (theory  of  radiation)  ). 

(4)  The  determination  of  the  chemical  composition  of  the 
mixture  in  equilibrium. 

(a)  The  streaming  method,  due  to  Deville  (Ann.  Chi  in.  Phann., 

135,  94,  1865),  who  was  able  to 

•  j  detect  the  dissociation  of  steam 

~~" i and  carbon  dioxide  at  very  high 

FIG  6?  temperatures    by   means    of   his 

so-called  "  cold-hot  tube  "  (tube 

chand  et  froid).  A  porcelain  tube  having  a  narrow  silver  tube 
running  along  its  axis  was  heated  to  whiteness  in  a  furnace, 
and  a  stream  of  cold  water  was  sent  through  the  silver  tube.  If 
steam  was  passed  into  the  space  between  the  two  tubes,  it  was 
partially  decomposed,  but  if  the  equilibrium  mixture  of  gases 
produced  in  contact  with  the  hot  wall  had  been  allowed  to  cool 
slowly  to  the  ordinary  temperature,  the  hydrogen  and  oxygen 
would  have  recombined.  If,  however,  the  mixture  strikes  against 
the  cold  silver  tube,  its  temperature  is  at  once  reduced  so  far 
that  the  region  of  false  equilibrium  is  reached,  the  equilibrium 
is  "  frozen,"  and  the  issuing  gas  can  then  be  analysed. 

The  same  result  might  be  achieved  by  taking  one  long  tube 
ac,  heating  up  a  length  ab  in  a  furnace,  and  cooling  the  part  be. 
Gas  is  then  passed  in  at  a,  and  attains  the  equilibrium  state  in 
ab.  The  equilibrium  is  "  frozen  ".in  be,  and  the  gas  issuing  from 
*  c  is  the  equilibrium  mixture  corresponding  with  the  temperature 
of  ab.  The  method  can  be  successfully  utilised  only  if  ab  is  long 
enough  to  bring  the  mixture  to  equilibrium,  and  be  short  enough 
to  freeze  the  equilibrium  before  back  action  sets  in. 

This  method  has  been  used  to  determine  the  dissociation  of 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    355 

steam  and  carbon  dioxide  by  von  Wartenberg  and  Nernst,  who 
employed  a  porcelain  pipette  heated  electrically  in  a  platinum 
tube.  The  bulb  was  7'5  cm.  long  and  2  cm.  diameter,  with 
an  outlet  capillary  tube  of  0'5  mm.  bore,  and  an  inlet  tube 
of  6  mm.  bore,  through  which  a  thermo-elenient  was  intro- 
duced. The  recombination  on  cooling  was  avoided  by  using 
a  rapid  current  of  gas  and  the  narrow  outlet  tube.  With  C02, 
even  when  so  rapidly  cooled,  satisfactory  results  could  only  be 
obtained  when  the  gas  was  perfectly  dry,  small  traces  of  moisture, 
as  is  well  known,  accelerating  the  reaction  enormously. 

(6)  The  Explosion  Method,  used  by  Horstmann.  The  mixture 
is  brought  to  a  very  high  temperature  by  firing  with  an  electric 
spark,  and  then  cools  very  rapidly.  The  temperature  at  which 
equilibrium  is  attained,  assumed  to  be  the  highest  temperature 
reached,  may  be  calculated  from  the  heat  of  reaction,  the  com- 
position of  the  mixture,  and  the  specific  heats  of  its  constituents. 
If  T  is  the  heat  capacity  of  the  mixture,  and  Q  the  amount  of 
heat  evolved  when  the  reaction  takes  place,  and  the  products  cool 
rapidly  to  TO,  the  temperature  of  reaction  T  is  given  by  : 

vr 

tt/T. 


Q=   I  Hi 
J  TO 


In  this  category  may  be  included  the  equilibria  attained  at 
high  temperatures  in  freely-burning  flames.  The  temperature 
of  an  oxy-hydrogen  flame  is  about  8500°  C.,  when  30  per  cent,  of 
the  steam  is  dissociated.  An  acetylene  flame  is  very  much 
hotter,  since  the  heat  of  dissociation  of  the  eudotherniic  CoH-j  is 
added  to  the  heat  of  combustion.  The  temperatures  attained  by 
explosion  in  enclosed  bombs  are,  however,  much  higher  than 
those  in  free  flames,  because  the  specific  heats  of  the  gases  at 
constant  volume  are  less  than  those  at  constant  pressure,  there 
is  less  cooling,  and  less  dissociation.  Temperatures  of  4500°  C. 
can  be  obtained  in  this  way,  and  M.  Pier,  who  has  recently 
elaborated  the  explosion  method,  has  been  able  to  measure 
equilibria  up  to  3300°  C.  There  seems  to  be  some  uncertainty 
in  these  measurements  as  to  the  cooling  correction  (cf.  Bjerriim, 
Zeitsckr.  physik.  Chem.,  79,  513,  1912). 

(c)  Acceleration  by  Catalysts. — The  reaction  is  allowed  to  take 
place  on  the  surface  of  a  solid  catalyst  heated  to  the  temperature 
at  which  the  equilibrium  is  to  be  measured.  The  gas  mixture 


356  THERMODYNAMICS 

in  contact  with  the  catalyst  rapidly  attains  a  state  of  equilibrium, 
and  then  diffuses  from  the  hot  surface  into  the  cold  surrounding 
gas.  The  equilibrium  is  then  "  frozen,"  in  the  same  way  as  in 
Deville's  method,  and  after  a  certain  time  the  gas  mixture  attains 
a  composition  corresponding  with  the  equilibrium  state  at  the 
temperature  of  the  hot  surface. 

Thus  if  a  platinum  wire  is  heated  electrically  in  a  flask  of 
steam,  the  latter  comes  to  the  equilibrium  composition 

2H.20  ^  2H2  +  02 
at  the  temperature  of  the  wire. 

This  method  is  due  to  Langmuir.  Measurements  have  also 
been  made  by  von  Wartenberg  and  Nernst,  and  Holt.  The  tem- 
perature of  the  wire  is  determined  from  its  resistance.  Pring 
has  investigated  the  mixture  of  hydrocarbons  produced  when  a 
carbon  rod  is  heated  electrically  in  a  globe  filled  with  hydrogen. 
The  temperature  of  the  rod  was  determined  by  the  optical  method. 
Haber  and  van  Oordt  have  measured  the  ammonia  equilibrium : 

2NH3  ^±  N2  +  3H2 
in  contact  with  iron,  cobalt,  etc.,  as  catalysts. 

(d)  Semipcrmeable  membranes. — If  a  gas  which  on  dissociation 
.  produces     hydrogen     is 

streamed  through  a  tube 
heated    at    an    assigned 

>n  \u  ?  ?/Hht-rnrP"ir  ~  'em?erature  and  T' 

\  A  A  /,  A    \  I    L_  taming  a  vacuous  palla- 

I    ''/  V  '•/  V  V  V     j--*  ^      J  dium     bulb      connected 

with   a   manometer,  the 
hydrogen  will  enter  the 

bulb  and  finally  attain  a  pressure  equal  to  its  partial  pressure  in 
the  mixture.  The  temperature  of  the  bulb  may  be  determined 
by  a  thermocouple,  or  optically. 

The  method  has  been  used  by  Lowenstein  with  H20,  HC1,  H2S, 
and  by  Vogel  von  Falckenstein  with  HC1,  HBr,  and  HI  (cf .  tables 
in  §  149). 

(?)  Dens/fy/.— The  methods  of  Dumas  and  Victor  Meyer  for  the 
determination  of  vapour  density  may  be  extended  for  use  at  high 
temperatures  by  forming  the  apparatus  of  refractory  material. 
Thus  Debray  used  the  method  of  Dumas  at  the  temperature  of 
boiling  zinc  by  making  the  bulbs  of  glazed  porcelain  instead  of 
glass.  Victor  Meyer's  method  was  extended  to  high  temperatures 


CHEMICAL  EQUILIBRIUM  IN  GASEOUS  SYSTEMS    357 

by  its  discoverer  in  the  same  way,  and  was  used  by  Meier  and 
Crafts  to  determine  the  vapour  densities  of  chlorine,  bromine, 
and  iodine  at  very  high  temperatures.  Nernst  constructed  a 
small  Victor  Meyer  apparatus  from  iridium,  heated  it  in  an 
electric  furnace,  and  measured  the  expansion  by  the  movement 
of  a  drop  of  mercury  in  a  horizontal  side  tube.  The  apparatus 
was  filled  with  an  inert  gas,  and  a  small  amount  of  the  substance, 
weighed  into  a  tiny  iridium  tube  by  means  of  a  quartz  micro- 
balance,  was  dropped  in.  The  temperature  was  measured 
optically.  In  this  way  the  dissociation  : 

Sa  ^  2S 

was  investigated,  and  the  vapour. densities  of  several  metals  were 
determined  by  von  Wartenberg. 

Preuner  and  Schupp  have  made  vapour-density  measurements 
with  sulphur  enclosed  in  a  quartz  bulb  heated  electrically  and 
connected  with  a  manometer  consisting  of  a  spiral  of  silica  tubing 
attached  to  a  small  mirror.  The  pressure  was  measured  by  the 
amount  of  unwinding  of  the  spiral.  The  same  method  has  been 
used  by  Bodenstein  and  Katavama  in  studying  the  equilibrium  : 
H2S04"—  H20  +  S03 

(/)  Electrochemical  Method. — In  this  the  value  of  the  equili- 
brium constant  K  is  calculated  from  the  maximum  work 
measured  by  means  of  the  electromotive  force  of  a  voltaic  cell 
(cf.  Chap.  XVI.). 

Further  particulars  of  these  methods  will  be  found  in :  W. 
Nernst,  Applications  of  Thermodynamics  to  Chemistry,  190(3 ; 
F.  Haber,  Thermodynamics  of  Technical  Gas  Reactions,  2nd 
edit.,  trans.  Lamb,  1909. 


CHAPTER   XIII 

EQUILIBRIUM    IN    DILUTE    SOLUTIONS 

155.     Chemical  Potentials. 

IN  the  section  on  chemical  equilibrium  in  gases  we  introduced  a 
magnitude  called  the  molecular  chemical  potential  of  a  component  : 

8U\  , 

•    •    • 


where  8»,  is  the  number  of  mols  introduced  into  the  mixture, 
whilst  the  magnitude  indicated  by  the  suffixes,  together  with  the 
masses  of  all  the  other  components,  remain  constant  ;  and  d<t>,  dV, 
rfU,  denote  the  resulting  increase  of  potential,  free  energy,  or 
intrinsic  energy,  respectively. 

The  condition  of  equilibrium  of  the  mixture  was  written  in  the 
form  : 

/M»i  +  prf>n<i  +  .  .  +  pfi»i  =  2/Z;§H(  =  0 
or  if  we  put 

bni  :  8«2  :  .  .  :  8??^  =  vi  :  v%  :  .  .  :  v{ 

2ft*4  =  0     ......     (2) 

This  notation  admits  of  generalisation  (Gibbs,  1876).  The 
total  energy  of  a  homogeneous  fluid  is  a  continuous  and 
single-  valued  function  of  the  masses  nil,  m^,  m3,  .  .  m-t,  of  its 
constituents,  of  the  total  volume  V,  and  the  total  entropy  S  : 

U    =    U(M»1./H2,     .     .     W;,V,S)  .  •  •         (3) 


,.         .     (4) 

/S,-»  (»l;V,S 

in  which  the  suffixes  denote  that 

V  and  all  the  masses  are  constant  in  forming  ?U/3S 
S       „          „          „          „          „  „      ?)U/8V 

V,  S  and  all  the  masses  except  w;  are  con- 
stant in  forming  BU/9»ij 

B  .  an     rp   au 

But8S  =  T;9T=-^ 
and  if  we  put  ^r~  =  fr  -    (5) 


EQUILIBRIUM  IN  DILUTE   SOLUTIONS          359 

where  /^  is  defined  as  the  chemical  potential  of  the  /-th  component  : 
rfU  =  TrfS  —  pd\  +  2/i.rf/n,      .         .         .     (6) 
If  we  now  put  : 

*  =  U  -  TS  +  ;>V 

*  =  U  -  TS 
we  have  : 

rf*  =  —  SdT  +  Vrfp  +  2/ 

d*    =    —    Sf/T    —  p(1\   +    2/A; 


The  conditions  of  equilibrium  (§§  51,  52) 

(f/U)S)V  =  0;  (d*)^  =  0;  (fWr.v  =  0     .         .     (8) 
all  lead  to  the  same  general  equation  : 

2/v7/M;  =  0         .         .         .         .     (9) 

If  instead  of  the  masses  (/«)  we  had  taken  the  numbers  of 
rnols  (H)  of  the  components,  together  with  V  and  S,  as  the 
independent  variables,  we  should  have  found  : 

2/ijdH;  =  0    .....     (10) 
or   2/ifi',-     =0    .         .          .         .         .     (11) 
where  /*,-  is  defined  by  (1). 

Since  (hi;  =  -^  where  M(-  =  molecular  weight,  we  have 
JVlf 

ji;   =  M,/i,       ......         (12) 

Illustration.  —  If  we  have  a  saturated  solution  of  a  salt  in  contact  with  solid 
crystals  of  salt,  the  whole  is  in  equilibrium  at  an  assigned  temperature  and 
pressure.  If  <p',  $"  are  the  thennodynamic  potentials  of  the  solution  and 
crystals  respectively,  and  if  we  suppose  a  further  very  small  mass  of  salt, 
8m,  to  pass  into  solution,  there  will  be  : 

an  increase  of  the  potential  of  the  solution  =  +  ;  —  S//i, 

a  decrease  of  the  potential  of  the  solid  =  --  -  5/». 

cm 
If  </>  is  the  total  potential  of  the  salt  and  solution  together  : 

t>  =<*>'  +  <*>"• 
The  condition  of  equilibrium  : 

(«*)!>  =  0 
requires  that  5<j>'  +  5<f>"  =  0 


/<y  _  <vrw  = 

\ciw       cm*/ 


860  THERMODYNAMICS 

Bat  jr-  —  //,  ~—  =  M",  the  chemical  potentials  of  the  salt  in  both  phases, 

and  hence  these  are  equal : 

M'  =  M" 

so  that  any  small  isothennal-isopiestic  change  from  the  equilibrium  state 
raises  the  potential  of  one  phase  to  the  same  extent  as  it  lowers  that  of  the 
other,  the  whole  potential  of  the  system  remaining  unchanged  to  the  first 
order. 


156.     Euler's  Theorem  on  Homogeneous  Functions. 

A  function  <f>(xi,xz,  .  .  #»),  of  n  variables  of  xi,  x*,  .  .,  xn  is  said 
to  be  homogeneous  and  of  the  m-tli  degree  with  respect  to  these 
variables  when  the  identity  : 

(f>(kxi,kxz,  .  .  ,  kxn)  =  km<j)(xi,x2,  .  .  ,  xn)      .         .     (1) 
is  verified  for  all  values  of  xi,  x%,  .  .  xn  and  k. 
Thus  $  [k(xi  +  8^1),  kx2,  .  .  ,  kxn)  =  km<j>(xi  +   tei,  x2,  .  .   ,  xtl) 


or,  proceeding  to  the  limit,  bxi  —  0  : 

.  ,  kxn)  _  ,  w_j  8^(a?i,      ,  .  .  ,    n  , 


Let  us  now  differentiate  both  sides  of  (1)  with  respect  to  k.     If 
we  put  kxi  =  HI  etc.,  we  have,  if  </>  =  $(k.i\,  kx2,  .  .  ,  kxtl)  : 

d<f)  dnn 
" 


.    .  ,  kxn)  ^(kxj,  kx2,  .  .  kxn) 

~" 


Hence,  from  (2)  : 

fyfo,  a:2,  .  .  ,    .r,,)  80(a-i,  a-2,  .  .     ,.r,() 


=  m<j>(xi,  x2,  .  .  ,  xn)  .  .         .     (3) 

which  is  a  very  important  theorem  due  to  Euler, 


EQUILIBRIUM  IN  DILUTE    SOLUTIONS          361 

157.     Potential  of  a  Solution. 

The  potential  of  a  solution  is 

$>!,  1Ha,  '"3,    •    •    ,  «!,-,   P,  T)        .  (1) 

where  in,-  is  the  mass  of  the  i-th  component.  If  all  the  masses 
are  increased  in  the  same  ratio  /.-,  the  potential  of  the  resulting 
mixture  is  /,-  times  that  of  the  first  : 

</»(A-W!,  A-/HJ,  .  .  ,  hnlt.  p.  T)  =  kftmi,  nig,  .  -  ,  m,,  P,  T)  .  (2) 
/.  <f>  is  a  homogeneous  function  of  the  first  degree  with  respect 
to  the  masses  (not  necessarily  linear),  hence,  from  Euler's 
theorem,  we  find  : 


$  (3) 

/H2  m, 

But  £±-  =  fix,  J£-  =  pa,  etc.,  the  chemical  potentials, 


Similarly,  with  the  variables  HI,  H2,  .  .  ,  H,,  p,  T(§  155)  : 

/ti"i  +  A*2»2  +  .  .  .  +  /t,-iii  =  </>  .         .  (4«) 

If  we  put  /HI  =  1,  ;H2  =  w*3  =  .  .  =»},-  =  0,  we  have 

«^  =  Mi      .....     (5) 

so  that  the  chemical  potential  of  a  pure  substance  is  its  therino- 
dynamic  potential  per  unit  mass. 

From  (2)  of  the  preceding  section  we  see  that  the  chemical 
potentials  are  homogeneous  functions  of  zero  degree  with 
respect  to  the  masses,  hence  from  Euler's  theorem  : 


,Ml   —  +   /H2  -£!-  +    .    .    .    +    ,H,  ^=-= 

Again,  since  d<f>  is  a  perfect  differential : 

g^ 


.     (6) 


362  THERMODYNAMICS 

and  generally  ~—  —  =  --''    (reciprocal  relation)  .       .     (7) 


-     (8) 


oni,  onii  onii 

A  very  useful  special  case  is  that  of  a  binary  mi.iiiin1 : 


The  chemical  potentials  depend  on  the  two  masses  only  through 
their  ratio  : 

s  =.  m2/nii 

.'.     (h)ii  —  —    —  r/-s  ;  ^"'2  — 

S 

and  it  is  easily  shown  that  : 


The  stability  of  the  equilibrium  of  the  solution  requires  that, 
for  all  isothermal-isopiestic  changes  : 

82(/>  >  0  .....     (10) 

But  ty  =  ?t  8m,  +  |^  8Wfl  +  ^  lp  +  ^  8T 
?/MI  9;»2  3;>  9* 

.'.  if  5j>  =  0,  6T  =  0, 


and  #     =          S/»i2  +  2         - 


>  0 
But 


8        = 

9s          9s 

>  0 


EQUILIBRIUM  IN   DILUTE    SOLUTIONS          363 

*j?  (Sw2  —  s&mtf  >  0 

.'.  2^  <  0.  and  ^2  >0    .  ,     (11) 

cs  cs 

This  simply  states  that  the  addition  of  a  further  quantity  of  the  second 
component  to  the  solution  increases  its  own  chemical  potential  and 
decreases  that  of  the  first  component,  when  the  solution  is  in  stable 
equilibrium. 

In  the  present  state  of  thermodynamics  the  calculation  of  the 
chemical  potential  of  a  component  of  a  solution  can  be  effected 
explicitly  in  two  cases  only  : 

(1)  Mixtures  of  gases  (§  143). 

(2)  Dilute  solutions. 

In  the  former  case  we  have  obtained  the  expression  : 

pi  =  T(?l  +  R/TU-,), 

and  we  shall  in  the  next  section  show  that  a  similar  expression 
holds  for  the  chemical  potential  of  a  component  of  a  dilute 
solution. 

158.     Equilibrium  in  Dilute  Solutions. 

The  determination  of  the  chemical  potentials  of  the  components 
of  a  solution,  in  terms  of  p,  T  and  the  masses  (or  concentrations) 
is,  from  what  precedes,  equivalent  to  finding  the  conditions  of 
equilibrium. 

The  form  of  the  chemical  potential  of  a  substance  present  in 
very  small  amount  in  a  solution  was  shown  by  Gibbs  as  early  as 
1876  to  be  a  logarithmic  function  of  the  concentration : 
li  —  A  +  B/HS. 

The  chemical  potential  of  each  component  is  known  when  the 
total  potential  4>  of  the  solution  is  determined  as  a  function  of 
the  composition. 

The  potential  *  of  a  solution  formed  of  /i0,  MI,  "-2,  •  •  mols  of 
substances  of  molecular  weights  MO,  >«i,  w-a,  ...  at  the  temperature 
T  and  pressure  p  will  be  a  function  of  p,  T,  and  the  «'s,  and 
the  total  energy,  entropy,  and  volume  are  functions  of  the  same 
variables : 

If  all  the  n's  are  increased  in  the  same  ratio,  U  and  V  are 
also  increased  in  the  same  ratio,  and  are  therefore  homogeneous 
functions  of  the  first  degree  in  those  variables.  U/HO  and 


864  THERMODYNAMICS 

are  therefore  unchanged  when  all  the  w's  are  changed  in  the 
same  ratio,  and  depend  on  the  w's  only  through  their  ratios, 

—  —  .  .  .  ;  and  if  the  values  of  n\.  Wa,  .  .  are  small  compared  with 
wo'  w0 

WQ,  and  U/MO,  Y/w0  are  finite,  continuous  and  differentiate  func- 
tions of  these  variables  they  are  necessarily  linear  (Taylor's 
theorem)  : 


»0  HQ  HQ 

I=  ,.„  +  ,,!'  +,.,»*  +  ......    (8) 

wo  "o  "o 

where  the  coefficients  u0,  wi>  •  •  ro,  ri,  •  •  are  independent  of  the 
w's  and  depend  only  on  the  temperature,  the  pressure,  and  the 
composition  of  the  components. 

If  we  put  HI  —  w2  =  .  .  =0  we  find  that  u0,  r0  depend  only  on 
the  properties  of  the  solvent  ;  but  HI,  TI  depend  on  the  properties 
both  of  the  first  solute  and  of  the  solvent,  but  do  not  depend  on 
the  other  solutes,  and  so  on.  There  is  therefore  an  interaction 
between  the  molecules  of  the  solvent  and  those  of  the  solutes,  but 
not  between  the  latter  themselves  (cf.  §  129).  When  the  solution 
is  more  concentrated,  the  interaction  between  solute  molecules 
may  be  included  by  taking  terms  of  higher  order  in  the  Taylor's 

series,  viz.,  such  terms  as   HH(-)  ,  and  2«12    -^-f-.  un  and  ui2 


may  be   called  coefficients  of   self,  and   of   mutual,  interaction 
respectively. 

If  we  write  (2)  and  (3)  in  the  form  : 

U  =  HQllo  +   »1»1  +  »2»2  +     ....  )  ,£. 

V   =  >WO    +    >hri  -)-  H3f3   -)-•••       ' 

we  see  that  they  are  equivalent  to  the  two  conditions  (1)  and  (2) 
of  §  128,  viz.  : 

(1)  The  change  of  total  volume  on  diluting  a  dilute  solution 
is  zero. 

For  if  the  solution  and  a  mol  of  pure  solvent  are  separate  the 
total  volume  is  »0r0  -f-  n^i'i  -)-..+  r0,  i.e.,  V  +  i'0,  whilst  if 
they  are  mixed  it  is  V  =  (??0  +  I)r0  +  »iri  +  •  •  >  »'.<•.,  the  same 
as  that  of  the  initial  system. 

(2)  The  heat  absorbed  on  diluting  a  dilute  solution  is  zero. 
For  the  intrinsic  energy  of  the  solution  and  mol  of  solvent 


EQUILIBRIA!  IN  DILUTE   SOLUTIONS          365 

separately  is  U  +  u0,  and  that  of  the  solution  U'  is  (n  +  l)"o 
+  «i"i  +  •  •  >  i-e->  aJso  U  +  HO- 

The  heat  effect  on  mixing  is  : 
W  -  W  =  U'  -  (U0  +  «o)  +  P  [V  -  (V  +  r0)]   =  0. 

These  results  assume  that  the  other  it's  remain  unchanged,  i.e., 
no  chemical  action  occurs  on  dilution. 

We  now  proceed  to  find  an  expression  for  the  entmpy  of  the 
solution.  If  the  composition  remains  constant,  the  solution  may 
be  treated  as  a  simple  fluid,  the  entropy  of  which  is  defined 
by  the  equation  : 

.K  =  '<U+rfY      .....     (5) 


...  rfg  =  „,      oo  +  ,h  +  „, 

The  «'s  and  U'B  depend  only  on  p  and  T,  but  not  on  the  w's, 
hence  the  coefficients  of  the  latter  must  be  perfect  differentials, 
so  that  there  must  exist  a  number  of  functions  s0,  *i,  *2>  .  .  .  of  p 
and  T  such  that  : 

7         du.2-\-  pdi:2  _. 

--     -- 


0  -  -  m  --  1  --  m  --  a  -    --  m  --         •     • 

The  integration  of  (6)  now  gives  : 

S  =  HO*O  +  "1*1  +  «a*2  +  •  •  +  C  .  .  (8) 
where  C  is  independent  of  p  and  T,  but  contains  that  part  of  the 
expression  for  the  entropy  which  depends  011  the  variability  of 
composition,  i.e.,  is  a  function  of  the  /i's.  To  find  C  we  imagine 
the  solution  converted  continuously  (i.e.,  without  separating  into 
various  phases)  by  suitable  alterations  of  p  and  T  but  constant 
w's  into  an  ideal  gas  mixture,  the  entropy  of  which  is  known,  by  the 
direct  separation  by  means  of  semipermeable  membranes  to  be  : 


This  imaginary  process  of  evaporation,  which  it  is  true  could 
only  proceed  as  a  succession  of  labile  states  analogous  to  the 
continuous  passage  of  a  liquid  into  vapour  along  the  James 
Thomson  isotherm,  (§  90)  is  legitimate  because  the  expression 
for  the  entropy  applies  to  all  states,  whether  states  of  equilibrium 
or  not,  and  the  w's  (together  with  p  and  T)  are  independent 
variables,  and  may  therefore  be  assumed  to  remain  constant. 

Since    expression    (8)   can,    by    change    of  p    and  T  alone, 
assume  the  form  of  (9)  only  if  : 
C  =  — 


366  THEEMODYNAMICS 

it  follows  that  the  entropy  of  the  mixture  must  be  : 

S  =  iioSo  +  niSi  +  .  .  —  K(«0^'o  +  n-ilnci  +  .  .)    .     (11) 
and  hence  the  potential  is  : 


.  .)]     +  X'Wfl   +    "lj'l   +    •  •) 

T*i  +  pci)  +  .  . 

-f-  KT  (iiolncQ  +  Wi^zt'i  -}-  •  •  ) 


=  T[/?o<po  +  »i9i  +  •  •  + 

=  TS^  (?i  +  K/wCf  )      .         .         .....     (12) 

where  : 

T<p(  =  ii;  -  Ts,  +  pi-i        .       '  .        .     (13) 
and  the  summation  extends  over  all  the  components. 

Tfo  +  R^)==|^=ft        .If.        .    (14) 

is  the  chemical  potential  of  the  i-th  component. 

Although  S  and  4>  are  homogeneous  functions  of  the  first  degree 
in  the  w's,  they  are  not  linear,  as  are  IT  and  V. 

The  condition  of  equilibrium  at  constant  temperature  and 
pressure  is  : 

8*  —  o,  when  8T  =  0  and  Sp  =  0, 
/.  Sw^i  +  Rliwt)  +  S(9/  +  K/wc,)8«(-  =  0 

As  in  the  case  of  a  gas  mixture  (§  142)  the  first  term  vanishes 
identically  : 

/.  S(9,-  +  Elnc^n-i  =  0          .         .         .     (15) 

It  may  be  shown  by  a  consideration  of  824>  that  this  is  positive, 
hence  the  equilibrium  is  stable  (§  157,  and  Duheni  :  Mecan.  chim., 
torn.  III.,  ch.  1  and  2). 

We  denote  a  system  composed  of  several  phases,  each  of  which 
is  a  dilute  solution,  by  the  symbol  : 

iiQiiio,  «i/»i,  ...  |  n0'm0',  ii'iri  ,  ...  |  HO'IHQ",  HI"  mi",  .  .  .  \ 

1st  Phase  2nd  Phase  3rd  Phase 

Special  cases  of  such  phases  are  ideal  gas  mixtures,  and  the 
limiting  case  of  a  pure  substance  in  any  state  of  aggregation. 
The  total  potential  of  the  system  is  : 

$  =  <f>  +</>'+  <£"  +  ... 
and  the  condition  of  equilibrium  at  constant  })  and  T  is  : 

0  =  8*  -  S<f>  4.  Sp  _|_  fy"  +   .  . 
i.e.  2(<po  +  'Rlnc0)8n0  +  (9!  +  Elnc^Bii!  +  .  .   =  0       .     (16) 


EQUILIBRIUM  IN  DILUTE   SOLUTIONS          367 

in  which   the  sign  2  indicates    summation    of   all    the   terms 
following  over  all  the  phases  of  the  system. 

As  before  we  may  replace  S»0,  S»i>  •  •  -  by  the  ratios  of  whole 
numbers  : 

SnQ  :  Bill  :  .  .  :  Sn0'  :  Siii  :  .  .  =  TO  :  TI  :  .  .  :  i'o  '•  v\   '•  •  • 
the  vs  being  positive  or  negative  according  as  the  substance  is 
produced  or  disappears. 

91  +  .  .  =  InK,,^  (17) 


where  K  depends  only  on  p  and  T,  but  not  on  the  composition. 
At  a  given  temperature  and  pressure  a  mixture  of  the  various 
molecular  species  settles  down  to  a  state  of  stable  chemical 
equilibrium  in  which  the  concentrations  in  each  phase  have 
finite,  although  in  some  cases  possibly  small,  values. 

The  late  of  mass-action  applies  to  each  phase  of  the  system. 

The  equilibria  in  the  various  phases  having  common  com- 
ponents are  not  independent,  but  must  be  related  by  definite 
ratios  of  the  concentrations  of  these  components. 

The  distribution  la  ic  applies  to  each  pair  of  phases  of  the  system. 

The  concentrations  are  contained  in  separate  terms,  and  the 
equation  (17)  for  the  whole  reaction  can  be  divided  into  a  number 
of  separate  equations  expressing  the  equilibrium  conditions  for 
various  possible  partial  reactions. 

The  equilibrium  is  established  as  though  each  partial  reaction  pro- 
ceeded separately,  and  the  other  components  containing  the  same 
elements  were  prevented  from  undergoing  change. 

The  deduction  adopted  is  due  to  M.  Planck  (Thermodynamik,  3  Aufl., 
Kap.  5),  and  depends  fundamentally  on  the  separation  of  the  gas  mixture, 
resulting  from  continuous  evaporation  of  the  solution,  into  its  constituents 
by  means  of  semipermeable  membranes.  Another  method,  depending  on 
such  a  separation  applied  directly  to  the  solution,  i.e.,  an  osmotic  process,  is 
due  to  vau't  Hoff,  who  arrived  at  the  laws  of  equilibrium  in  dilute  solution 
from  the  standpoint  of  osmotic  pressure.  The  applications  of  the  law  of 
mass-action  belong  to  treatises  on  chemical  statics  (cf.  Mellor,  Chemical 
Statics  and  Dynamics)  ;  we  shall  here  consider  only  one  or  two  cases  which 
serve  to  illustrate  some  fundamental  aspects  of  the  theory. 

159.     Influence  of  Temperature   and   Pressure   on  the   Equi- 
librium. 

For  each  phase  we  have  : 

co  -f-  vilnci  -f   •  •  •   =  --    (^o?o  +  ^i?!  +   •  •  )  =  /»K  .     (1) 


368  THERMODYNAMICS 


.    (dlnK\   _     _  d_ 

'  V  dp  )r~       dp"  R 

_         8     yp?o  +  PI<PI  +  •  •  • 

-    sf-       -R~ 


But  ^L  .     -    ;  ,     (4) 

/.  for  the  total  variation  of  9,  we  have  : 


But 


where  Av  =  VQVO  +  ^ii/-!  +  .  .  .  .         .        .     (8) 

is  the  totaZ  increase  of  volume  in  consequence  of  the  chemical 
change  ; 

and 


wvr'      ~w 

where  w  =  u  +  Pr 

and  Qp  =  v0ic0  +  viu-i  + (10) 

is  the  total  absorption  of  heat  at  constant  pressure  during  the 
reaction. 

Equations  (7)  and  (9)  contain  the  theory  of  the  influence  of 
pressure  and  temperature  on  the  equilibrium ;  they  are  identical 
with  those  deduced  for  gaseous  systems  (§  145). 

One  or  two  general  principles  follow  at  once  from  the  form  of 
the  expressions : 

(1)  If  Ar  —  0,  K  is  independent  of  p, 

if  A  r  >  0,  K  Decreases  when     increase8. 
1  increases 

(2)  If  Q,,  =  0,  K  is  independent  of  T, 

ifQp^O,K  iincreases   when  T  increases, 
(decreases 

We  observe  that  there  can  be  a  definite  state  of  equilibrium 


EQUILIBRIUM  IN  DILUTE   SOLUTIONS         369 

even  when  the  heat  of  reaction  is  zero,  in  direct  opposition  to 
Berthelot's  principle  (§  118)  which  makes  the  state  indefinite 
when  Q  =  0. 

(3)  If  any  concentration  vanishes,  c{  =  0,  the  expression  on 
the  left  of  (17),  §  158,  becomes  infinite,  since  lnct  =  InQ  —  —  GO  . 
But  ZwK  cannot  become  infinite  when  p  and  T  are  finite,  and 
hence  all  the  possible  molecular  species  must,  in  the  equilibrium 
state,  be  present  in  finite,  though  possibly  exceedingly  small 
amounts. 

Only  when  T  =  0  and  Q  is  finite  does  the  equation  (9)  show 
that  InK  is  infinite  and  one  or  more  of  the  c's  must  then  vanish, 
i.e.,  the  reaction  is  complete  in  the  sense  of  Berthelot's  rule. 

The  law  of  mass-action  of  Guldberg  and  Waage  is  therefore 
deducible  on  thermodynamic  grounds  for  mixtures  of  ideal 
gases,  and  for  ideal  dilute  solutions,  but  for  no  other  cases.  It  is 
necessary  to  emphasise  this  point  because  there  seems  to  be  a 
general  opinion  amongst  chemists  that  the  law  is  one  of  perfect 
generality,  and  that  its  application  to  every  possible  type  of 
chemical  equilibrium  is  therefore  allowable.  Whilst  in  an  em- 
pirical sense  this  may  be  true,  we  must  not  be  surprised  if 
divergencies  are  met  with  in  regions  which  lie  beyond  the  range 
of  application  sanctioned  by  the  assumptions  introduced  into  the 
derivation  of  the  law. 

160.    Examples   on   the    Application    of    the     Equations     of 
Equilibrium. 

(1)  Two  components  in  one  phase,  e.g.,  the  electrolytic  dissocia- 
tion of  a  salt  in  aqueous  solution  : 

CHsCOOH  =±  CH3COO  +  H 

A  large  number  of  other  reactions  are  also  possible,  e.#.,  hydra- 
tion  of  the  various  substances,  or  (if  the  solute  is  a  salt  of  a  weak 
acid  or  base)  hydrolysis,  but  in  all  cases  the  concentration  of 
each  molecular  species  is  defined  by  the  total  amount  of  solute 
in  a  given  mass  of  solution,  and  the  ionisation  proceeds  as  if  all 
the  other  reactions  did  not  occur  at  all  (cf.  §  158). 
The  system  is  :  + 

noHaO,  w^HaCOOH,  w2CH8COO,  n3H 
The  reaction  is : 

v  o  =  0,  z>i  =  —  I,v2  =  i>3  =  1. 


370  THERMODYNAMICS 

The  equilibrium  equation  is  therefore  : 

—  Inci  +  Inc2  +  /»c3  =  /nK 
or,  since  ^2  =  <"3  ; 

^  =  K 

Cl 

n0  wi  n2  "3 

where  e0  =  -,  Cl  =-  ,  c2  =-,  c3  =— 

n  =  HO  +  «i  +  HS  +  wg. 

Wi  -f-  Wo       7?i  -4-  no 
Put  ci  +  c2  =  e  =  —  ^—  =  —  —  —  approxi 

then  c  is  known  if  the  total  mass  of  solute  is  known  : 


K 


HI 


As  c  (i.e.,  the  dilution)  increases,  the  ratio  —  increases  to  a 

limiting  value  of  unity  for  complete  ionisation. 

The  transformation  of  numerical  concentrations  c,  to  volu- 
metric molecular  concentrations  £,  cannot  be  effected  in  the  case 
of  solutions  so  readily  as  with  ideal  gas  mixtures  (§  121),  on 
account  of  the  changes  of  density. 

We  may  put  pci  =  £lf  etc.,  approximately,  where  p  is  the 
density  of  the  solvent,  hence  : 


But  for  1  mol  originally  dissolved  : 

c        I  —  a      r  _a 

£1-^-,  &-T 

where  a  =  degree  of  ionisation,  V  =  total  volume, 


which  is  called  Ostwald's  Dilution  Law. 

The  measurements  of  a  by  means  of  the  electrical  conductivity 
show  that  the  dilution  law  holds  good  for  weak  electrolytes 
(a  small),  but  for  strong  electrolytes  (a  large)  it  fails  utterly. 
This  behaviour  has  given  rise  to  a  considerable  amount  of  dis- 
cussion, a  critical  review  of  which  will  be  found  in  a  paper  by 
the  author  ("  Ionic  Equilibrium  in  Solutions  of  Electrolytes  ")  in 
the  Trans.  Chem.  Soc.,  97,  1158,  1910.  It  appears  that  in  this 


EQUILIBRIUM  IN  DILUTE   SOLUTIONS         371 

case  the  ordinary  law  of  mass-action  fails,  as  it  usually  does  when 
electrical  energy  participates  in  establishing  a  state  of  chemical 
equilibrium  (formation  of  ammonia  by  the  electric  discharge, 
production  of  ozone,  etc.).  The  equation  proposed  by  Larmor  on 
the  basis  of  kinetic  molecular  theory : 


=  X 


(1  —  a)(V  +  fa) 

where  e,  \  are  constants,  has  been  shown  by  the  author  (loc.  cit.) 
to  give  quite  good  results,  and  reduces  to  Ostwald's  form 
when  a  is  small  (i.e.,  weak  electrolytes).  Thus  with  NaN03  (e  = 
23  X  105) : 


V 

a 

X  X  10* 

K'(0stwald)xl05 

107 

0-9836 

0-48 

0-58 

106 

0-9617 

0-74 

2-42 

167  X  103 

0-9295 

0-53 

7-34 

(2)  Equilibrium  of  a  gas  standing  over  its  saturated  solution  : 
HO»'O,  HiWi  |  ;?o'?»o' 
solution  gas. 


wo  +  »i 

Put  wii  =  anio,  then  the  passage  of  a  mol  of  dissolved  gas  into 
the  gas-space  is  represented  by : 

J'O  =r  0,    V\  =:    —   1,     VQ    =  Ct, 

and  the  equilibrium  equation  : 

reduces  to : 

—  Ind  =  /»KpiT 

The  solubility  ci  is  therefore  a  function  of  temperature  and 
pressure. 

In  this  case,  and  generally  where  gases,  as  distinguished  from 
liquids  and  solids,  participate  in  a  reaction,  the  dependence  on 
pressure  is  fairly  considerable : 

/3/»K\  _       Ar 

te/rJr'RT 

RT     , 


Now 


B  B  2 


372  THERMODYNAMICS 


— 

\  dp  /  T    \    p 
.-.  Jnt'i  =  alnp  +  const. 
or  cj  =  Cp*  (T  const.) 

so  that  the  solubility  of  the  gas  at   constant   temperature   is 
proportional  to  the  power  a  of  the  pressure. 

In  many  cases  it  is  found  that  a  =  1    /.  Ci  =  Cp,  which  is 
Henry's  law  (§  126). 

For  the  dependence  on  temperature  we  have  : 


But  Inci  =  InC  +  alnp 


. 

8T/p    RT2 
Qp  is  the  heat  absorbed  when  a  mol  of  gas  is  abstracted  from  a 

large  volume  of  saturated  solution,  and  the  solubility  /increases 

^decreases 
with  rise  of  temperature  according  as 

Q   •   (negative  (heat  evolved) 

p     [positive  (heat  absorbed). 

In  the  majority  of  cases  the  solubility  decreases  with  rise  of 
temperature,  indicating  that  heat  is  absorbed  when  the  gas  is 
abstracted  from  its  saturated  solution  ;  with  the  group  of  inactive 
gases,  according  to  Estreicher  (§  126),  the  opposite  effect  is 
observed. 

In  Planck's  investigation  of  equilibrium  in  dilute  solutions,  the 
law  of  Henry  follows  as  a  deduction,  whereas  in  van't  Hoff's 
theory,  based  on  the  laws  of  osmotic  pressure  (§  128),  it  must 
be  introduced  as  a  law  of  experience.  The  difference  lies 
in  the  fact  that  in  Planck's  method  the  solution  is  converted 
continuously  into  a  gas  mixture  of  known  potential,  whilst  in 
van't  Hoff's  method  it  stands  in  equilibrium  with  a  gas  of  known 
potential,  and  the  boundary  eondition  (Henry's  law)  must  be 
known  as  well.  Planck  (Tliermotii/nainik,  loc.  cit.)  also  deduces 
the  laws  of  osmotic  pressure  from  the  theory. 

In  the  same  way  we  can  investigate  the  equilibrium  between  a 
sparingly  soluble  liquid  or  solid  substance  and  its  solution 
(cf.  §  182).  The  change  of  molecular  state  which  sometimes 


EQUILIBRIUM  IN  DILUTE   SOLUTIONS         378 

occurs  when  a  substance  passes  into  solution  (association,  or  dis- 
sociation) can  also  be  expressed  by  the  appropriate  values  of  v. 
Thus,  in  the  system  composed  of  a  solution  of  a  sparingly  soluble 
binary  electrolyte  in  equilibrium  with  solid  : 


WH20,  »iAB,  »2A,  »3B   |   Ho'AB 
solution  solid 

we  may  divide  the  change  : 


into  the  two  partial  changes  : 

(a)  the  precipitation  of  a  mol  of  AB  : 

r0  =  0,  ri  =  —  1,  r2  =  0,   v3  =  0,  J'o  =  *-f  1 

(6)  the  dissociation  of  a  mol  of  AB  : 

j'0  =  0,   i'i  =  —  1,  ra  =  1,  r3  =  1,  VQ   =  0. 

Thence  :  —  Inci  =  InK       .         .         .         .     (a) 

—  /wci  +  lncz  +  Inc3  =  InK'       .         .         .     (6) 

(a}  shows  that  the  concentration  of  unionised  solute  is  definite 
at  a  given  temperature  and  pressure  ;  and  (b)  that  the  concentra- 
tion of  the  ionised  part  is  determined  from  that  of  the  unionised 
part  by  the  ordinary  equation  of  example  (1).  The  total  solute 
is  ci  +  c2  =  ci  +  c3,  and  is  known,  and  hence  K,  K'  are  deter- 
mined by  a  further  measurement  of  02  by  the  freezing-point  or 
conductivity.  If  the  two  measurements  are  repeated  at  another 
temperature,  the  temperature  coefficients  of  /wK  and  //iK'f  and 
hence  the  heats  of  solution  and  dissociation,  Q^  and  Q'^,  are  found. 
Thence  we  find  the  heat  absorbed  when  a  mol  of  electrolyte  is 
dissolved  in  sufficient  solvent  to  produce  a  saturated  solution  : 


If  we  integrate  the  equation  : 

'  _  Q'p 
~ 


over  a  small  range  of  T  on  the  assumption  that  Q'p  is  constant, 
we  find  : 


\TI     T2/ 

so  that   Q'p  is  determined  from  the  temperature  coefficient  of 
the  conductivity. 


374 


THEKMODYNAMICS 


The  equations  for  the  vapour  pressures  and  freezing-points  of 
dilute  solutions  are  also  readily  deduced  from  Planck's  equation. 

The  linear  relation  between  the  depression  of  freezing-point  and 
the  concentration  strictly  applies  only  to  infinite  dilution  although 
it  holds  good  approximately  up  to  decinormal  concentration. 

J.  B.  Goebel  (Zeitschr.  physik.  Cliem.,  53,  213,  1905 ;  54,  314, 
1906;  71,  652,  1910)  has  found  an  empirical  equation  for  the 
depression  of  freezing-point  in  aqueous  solutions  of  total  mole- 
cular concentration  £n : 

&  =  0-705  log  (1  +  A)  +  0-24  A  +  0'004  A2 

CANE  SUGAR. 


A 

t  calcd. 
sn 

t  obs. 

0-0532 
0-2372 
2-0897 

0-0286 
0-122 
0-864 

0-0283 
0-122 

0-837 

With  a  binary  electrolyte  : 

NaCl  z±  Na  +  Cl' 
6  & 

the  total  salt  concentration  £    =  £„  -f-  £2 
,%fw  =  li  +  26 

and  the  equation  of  mass-action  gives  : 


which  gives  K  in  terms  of  A. 

From  this  we  can  calculate  the  value  of 


for  a  given  concentration  of  solute,  £. 

Ternary  electrolytes  dissociate  in  two  stages  : 

(i.)  CaCl2  =  CaCl  +  Cl 

&  &        & 

(ii.)  CaCl  =    Ca  +  Cl 


EQUILIBRIUM  IN   DILUTE    SOLUTIONS          375 
and  we  have  the  relations  : 


&  =  &  +  2&  =  &  -  £ 

To  each  equilibrium  there  corresponds  an  equation  of  mass- 
action  : 

£a£a  _  V  .  £a£*  _  T- 
—  f  --  JM  j  -?  --  Av2. 

Cl  C3 

The  determination  of  the  state  of  the  dissolved  solute  is  pos- 
sible if  two  measurements  are  made,  for  if  we  denote  all  the 
above  magnitudes  by  dashed  symbols  for  the  second  solution  : 


&=• 


«&=*=& 


K!  K-2 

CaCl2  5-91  0'109 

H2S04  0-45  0-017-2 

The  second  dissociation  therefore  proceeds  only  to  a  very  slight 
extent  in  comparison  with  the  first. 

The  theoretical  treatment  of  the  dissociation  of  electrolytes  has  been 
extended  by  Jahn  and  by  Xernst,  who  have  introduced  assumptions. 
equivalent  to  replacing  the  linear  expressions  for  the  energy  and  volume 
given  by  Planck  by  expressions  containing  higher  powers  in  the  Taylor's 
series  and  supposed  to  take  account  of  the  self  and  mutual  interactions 
between  the  solute  molecules.  The  resulting  equations  are  so  extraordinarily 
clumsy,  and  difficult  of  application,  as  to  prohibit  anything  more  than  a 
reference  to  them  here,  and  in  addition  their  physical  significance  is  far  from 
clear.  (Cf.  Partington,  Tram.  Chem.  Soc.,  97,  1158,  1910.) 


161.     Equilibria  in  Heterogeneous  Systems. 

We  shall  consider  the  equilibria  established  in  systems  com- 
posed of  various  solid  phases,  each  of  which  is  a  pure  substance 
(i.e.  not  a  solution)  in  contact  with  a  gaseous  phase. 


376  THERMODYNAMICS 

The  potential  of  the  system  is  the  sum  of  the  potentials  of  the 
gas  and  of  the  various  solids  : 

<t>=<j>  +  n'<l>'(p,T)+n"<l>"(p,rF)+   .  ,  =<j>  +  2n'p  .  .  (1) 

The  change  of  potential  for  a  small  isopiestic-isothermal 
change  : 

Bn'  :  Bn"  :  .  .  :  Bnj.  :  8;ia  :  .  .  =  v'  :  v"  :  .  .  :  v:  :  va  :   .  . 

is  S<i>  =  fy  +  <j>'bnr  +  </>"5n"  +  .  .  . 
=  T  [(91  +  RZwei)8rti  +  (92  +  RZ»c2)6n2  +  .  .  ]  +  2<f>'8n'    .  .    (2) 

If  8*  >  0  the  process  is  impossible  ;  if  S<I>  <  0  the  process  is 
possible  and  irreversible  ;  if  S<$>  =  0  the  process  is  reversible  and 
the  system  is  in  equilibrium  : 

+  •  •  ]  +  £</>'&<'  =  0 

^'292  +  •  •  )  +  -^" 


+  j^Zwca  +..=  —  -  (1/191  +  ^92  +   .  . 
ri 

+  Sw'9')  —  InK     .         .         .     (3) 

where  K  depends  on  p  and  T   alone,  and  is  the  equilibrium 
constant. 

Example.  —  The  action  of  steam  on  red-hot  iron  : 

3Fe  +  4H20  ^  Fe304  +  4H2 
v'  =  -  3,  v"  —  +  1,  v'"  =   .  .   =  0  ;  vi  =  —  4,  vz  =  +  4, 

va=  .  .  =0 

p(492  —  491  +  9"  —  89')  =  ?«K 

•**-« 


or  ~=  \,/  K  =  const. 

Partial  pressures  may  be  substituted  for  concentrations  (§  122). 

We  see  that  in  determining  the  equilibrium  the  concentrations 
of  the  solids  do  not  appear  at  all.  This  important  result  was 
first  stated  by  Guldberg  and  Waage  in  1867  in  the  form  that 
"  the  active  mass  of  a  solid  is  constant."  It  is  true  only  when 
the  solids  are  of  unvarying  composition. 

If  the  coexisting  liquid  or  solid  phases  are  not  pure,  but  solu- 
tions, equilibrium  will  be  established  when  the  chemical  potential 


EQUILIBRIUM  IN  DILUTE   SOLUTIONS         377 

of  every  component  is  the  same  in  all  phases  which  contain  it  as 
an  actual  component.  If  any  phases  are  free  from  any  com- 
ponent, its  chemical  potential  in  the  other  phases  must  every- 
where be  less  than  the  values  it  would  have  if  present  in  those 
phases  in  infinitesimal  amount.  For  the  change  of  thermo- 
dynamic  potential  when  a  mass  bm  of  the  component  passes  from 
a  phase  containing  it  at  chemical  potential  p  to  one  containing 
it  in  infinitesimal  amount  at  chemical  potential  p^  is  S</>  = 

(fiQ  —  /A)  SHI,  and  for  equilibrium  S</>  _  0  .'.  /JLO  >  /i.    The  change 

in  question  is  unilateral,  i.e.  the  opposite  change  is  excluded  by 
the  absence  of  the  component  from  the  second  phase. 

When  the  phases  are  dilute  solutions  the  general  equation 
deduced  at  the  beginning  of  the  section  may  be  applied. 

Example.  Dissociation  of  fused  cuprous  oxide  : 


the  copper  dissolving  in  the  cuprous  oxide  to  form  a  solution 
increasing  in  concentration  as  dissociation  proceeds.  The  system 
is: 

HoCUgO,  WiCu   I    H</02 

liquid          gas 

The  change  is  :  VQ  =  —  2,  vi  =  -j-  4,  VQ  —  +  1 
/.  —  2  Inc0  +  4  Inci  +  Inc0'  =  IriK. 

Inc0  =  In  —^  —  ;  lnCl  =  In  -^  -  ;  Inc0f  =  In  ^  =  0 
wo  +  Wi  n0  +  wi  wo 


Co 


This  shows  that  the  pressure  is  not  a  function  of  the  tempera- 
ture alone,  but  depends  also  on  the  relative  proportions  of  the 
two  substances  in  the  liquid  phase. 

162.     Effect  of  Temperature  and   Pressure   on   Heterogeneous 
Equilibria. 

With  our  previous  notation  : 


sz,  •  .  for  the  gaseous  phase, 


—  s",  .  .  .  for  the  condensed 


378  THEBMODYNAMICS 

phases,  we  have,  for  the  total  variations  with  temperature  and 
pressure  : 


/8<p'\  _       w'  +  pv'  _  _  w' 


RT  ~RT2- 

where  Ar  =  total  change  of  volume  in  the  reaction, 

Qp  =  heat  absorbed  in  the  reaction. 

The  contribution  of  the  condensed  phases  may  be  neglected  in 
considering  the  influence  of  pressure,  but  is  very  important  in 
considering  the  influence  of  temperature.  Thus,  in  §  119  we  found 
that  the  quantities  of  heat  evolved  in  the  reactions  : 

2H2  +  02  —  2H2  0 
8Fe  +  4H20  ^  Fe804  +  4H2 

approach  a  maximum  and  a  minimum  respectively  with  rise  of 
tempei'ature,  so  that  the  changes  of  the  equilibrium  constants  are 
opposite  in  sign,  although  the  reaction  in  the  gaseous  phase  is 
the  same  in  both  cases. 

The  equation  (2)  may  be  integrated  by  means  of  the  equation 
of  Kirchhoff  (§  58)  : 


Q  =  Qo  +     (IV  -  T,XT        .         .         ,         .     (8) 


EQUILIBRIUM  IN  DILUTE   SOLUTIONS         379 

Example  1.     Action  of  steam  on  red-hot  iron  : 
3Fe  +  4H20  ^  Fe3  04  -f  4H2 
The  substitution  f  —  T  =  a  -f  /3T  (§  147  (t-)) 

gives  :  InK  =  -Q^+^LlnT  +|.T  +  const. 

We  have  found  (§  119): 

Qo  =  -  39655  ;  a  =  -  13*71  ;  0  =  0'00656 


...  InK  =  _/nT  +  T  +  consfc. 

Kl  It  11 

Put  E  =  1'985,  and  transform  to  ordinary  logarithms  (R/nlO 
=  4-571): 


. 

log  K  =      p  -  2-999  log  T  +  0'00143T  +  (const.)' 

Deville  (1870)  found  K  =  0'048  when  T  =  473, 
/.  (const.)  '  =  9'303 

/.  log  K  =  ^P_  _  3  log  T  +  0-00143T  +  9'303. 

Example  2.     The  heat  of  hydration  of  a  salt  may  be  calculated 
from  the  change  of  dissociation  pressure  with  temperature  : 
BaCl2.  2  H20  =  BaCl2  +  2  H20 

vapour 


where  —  \  is  the  heat  evolved  in  the  formation  from  anhydrous 
salt  and  water  vapour.  The  heat  of  formation  from  liquid  water 
may  be  calculated,  on  the  assumption  that  water  vapour  is  an 
ideal  gas,  by  subtracting  from  A  the  heat  of  evaporation : 

din  1P-} 

A  _  -orpa  (  (UnP        !lln7M  —  -RTMJ        VTr/ 
\  dT    "    df~)  ~  ~dT~ 

where  ?r  =  vapour  pressure  of  water  at  T. 
This  equation  is  due  to  Horstmann  (1871). 


CHAPTEK  XIV 


GENERAL   THEORY   OF    MIXTURES    AND    SOLUTIONS 

163.     Mixed  Liquids. 

Of  great  practical  importance  are  the  liquid  solutions  produced 
by  mixing  together  pure  liquids.  The  simplest  type  comprises 
the  binary  liquid  mixtures,  such  as  mixtures  of  water  and  alcohol. 

If  such  a  mixture  is  distilled  in  a  retort,  the  distillate  contains, 
in  general,  the  two  substances  in  a  different  ratio  from  that  in 


"ft 


100  A 


100  B 


fOOA 


10 OB 


FIG.  69. 


FIG.  70. 


the  original  mixture,  and  this  ratio  alters  continuously  during 
the  process.  It  follows  that  the  composition  of  the  vapour  phase 
(which  on  condensation  yields  the  distillate)  is,  in  general, 
different  from  that  of  the  liquid  phase  with  which  it  is  in 
equilibrium ;  otherwise  the  whole  mixture  would  distil  with  un- 
changing composition. 

If  the  distillation  is  carried  out  at  constant  total  pressure  (e.tj., 
under  atmospheric  pressure)  there  will  be,  in  general,  a  change  of 
temperature,  for  if  it  were  possible  to  separate  a  mixture  into  two 
parts  of  different  composition  at  a  constant  temperature,  we 
could  remix  these  through  semipermeable  diaphragms  and  so 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    381 

obtain  a  finite  amount  of  work  in  an  isothermal  cycle,  which  is 
impossible. 

If,  however,  the  temperature  is  maintained  constant,  the 
pressure  exerted  by  the  vapour  will,  in  general,  change  during 
distillation,  for  if  the  liquid  could  be  separated  into  two  parts  of 
different  composition  at  a  fixed  temperature  and  pressure,  these 
could  be  remixed  with  performance  of  work  as  before. 

If  the  temperatures  corresponding  with  coexisting  compositions 
of  the  two  phases  at  constant  pressure  are  represented  on  a 
diagram,  one  obtains  two  curves  which  may  be  called  the  T -liquid 
and  T -vapour  curves  respectively. 

If  the  total  pressure  of  the  vapour  at  constant  temperature  is 
represented  as  a  function  of  the  compositions  of  the  two  phases, 
the  p-liquid  and  p-rapour  curves  are  obtained.  The  ^-liquid 
curves — that  is,  the  curves  representing  the  total  vapour  pressures 
of  liquid  binary  mixtures  as  functions  of  the  composition  of  the 
liquid  phase — are  most  important ;  they  are  usually  referred  to 
simply  as  "  the  vapour-pressure  curves  of  the  mixture."  Each 
curve  is  an  isotherm. 

The  points  a,  /3,  on  the  horizontal  lines,  represent  the  composi- 
tions of  liquid  and  vapour  phases  in  equilibrium  at  a  given  tem- 
perature or  pressure. 

164.     Classes  of  Binary  Mixtures. 

Three  classes  of  mixtures  may  be  distinguished : 

(1)  Immiscible  Liquids  (i.e., liquids  very  sparingly  soluble  in  each 
other,  such  as  water  with  carbon  disulphide),  in  which  the  total 
pressure  is  equal  to  the  sum  of  the  pressures  of  the  pure  liquids. 

(2)  Partially  Miscible  Liquids  (water  and  ether),  which  exhibit 
a  total  pressure  practically  equal  to  that  of  the  more  volatile 
component  (ether). 

(3)  Completely  Miscible  liquids  (benzene  and  alcohol)   show 
more  complicated  relations.     The  total  pressure  is  usually  inter- 
mediate between  the  pressures  of  the  pure  liquids,  and  in  many 
cases  less  than  the  sum  of  these,  but  greater  than  that  of  the 
most  volatile  component  (Regnault). 

The  vapour  pressure  relations  of  mixed  liquids  were  cleared  up 
experimentally  by  the  Russian  chemist,  Dmitri  Konowalow 
(1881) ;  the  theory  had  previously  (unknown  to  Konowalow)  been 
developed  by  J.  Willard  Gibbs  in  1875. 


382 


THERMODYNAMICS 


165.     Completely  Miscible  Liquids. 

Konowalow  distinguished  three  types  of  curves  of  total  pressure 
in  the  case  of  completely  miscible  liquids : 

(i.)  P  lies  between  pi  and  p%,  the  pressures  of  the  pure  com- 


Water  +  Methy/,  Alcohol 
FIG.  71. 


Water  +  Propyl  Alcohol 
FIG.  72. 


100° 


ponents  (methyl  alcohol  and  water,  acetic  and  propionic  acids 
and  water). 

(ii.)  P  exhibits  a  maximum  (water  with  propyl  and  isobutyl 
alcohols,  and  butyric  acid).  [The  case  of  ethyl  alcohol  and  water 
exhibits  a  maximum  pressure  with  96'0  per  cent,  of  alcohol.] 

(iii.)  P  exhibits  a  minimum  (water 
with  formic  acid). 

In  some  cases  the  curves  of  group  (1) 
showed  a  point  of  inflexion  (water  with 
methyl  alcohol). 

Konowalow's  curves  for  three  typical 
mixtures  are  shown  in  Figs.  71-73. 
Each  total  pressure  curve  may  be 
regarded  as  built  up  from  two  partial 
pressure  curves,  the  ordinate  of  every 
point  being  the  sum  of  the  partial 
pressures  over  the  liquid  having  the  composition  of  the 
abscissa. 

If  there  are  NI,  N2  mols  of  the  two  components,  I.  and  II.,  in 
the  liquid  at  any  instant,  the  ratios  : 


Water  +  Formic 
FIG.  73. 


are  denned  as  the  molecular  fractions  of  the  components,  respec- 
tively ;  they  are  identical  with  the  numerical  concentrations. 


GENEEAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    383 


166.     The  Partial  Pressure  Curves. 

Consider  the  two  limiting  solutions  of  I.  and  II.  : 
(a)  If  only   a   small   amount   of   II.   is   present,  the  vapour 
pressure  of  I.  will  be  lowered  in  accordance  with  Raoult's  law  : 

Pi  -.Pi      __N2 


N 


- 


.'.  pi  =xP,  ....     (1), 

where  p\,  PI  are  the  partial  pressure  of  I.  and  its  pressure  in  the 
pure  state,  respectively. 

The  right-hand  portion  of  the  curve  is  therefore  a  straight  line 
which  would,  if  produced,  pass 
through  zero  pressure  on  the  II. 
axis  (x  =  0).  The  rectilinear  rela- 
tion extends  up  to  the  limit  of 
validity  of  (1). 

(b)  If  only  a  small  amount  of  I. 
is  present,  its  partial  pressure  is 
given  by  Henry's  law,  because  we 
may  suppose  the  mixture  to  have 
been  produced  by  bringing  the 
vapour  of  I.  in  contact  with  liquid 
II.  Thus 

Pl=1TlPl       .  .        (2), 

where   TJI   is   a   constant    which    is 

inversely  proportional  to  the   solu- 

bility A  of  the  vapour  of  I.  in  liquid  II.     Equations  (1)  and  (2) 

are  identical  if  wi  =  PI. 

In  the  geometrical  representation  of  the  partial  pressures  of  L, 
we  have  on  the  right  the  straight  line  PiQ  corresponding  with 
(1),  and  on  the  left  one  or  other  of  the  three  initial  curves  Oa, 
Ob,  Oc,  corresponding  with  the  equa- 
tion TTi  =  PI  and  the  inequalities 
TTi  <  PI,  TI  >  PI,  respectively. 

The  complete  partial  pressure  curve 
will  therefore  be   one  of   the   three 
types  OcQPi,  OaQPi,  O&QPi,  which  may  be  denoted  by  c,  a,  b,  and 
called  neutral,  positive,  and  negative,  respectively. 

The  total  pressure  curve  will  be  made  up  additively  of  two 
such  curves,  the  possible  combinations  being  aa,  bb,  cc,  ab,  ac,  be. 


100% 

I 


FIG.  74. 


X 


FIG.  75. 


384  THERMODYNAMICS 

A  complete  set  of  diagrams  of  these  combinations  will  be  found 
in  Ostwald's  "Lehrbuch,"  II.,  2,  (1),  618—626;  the  cases  where 
the  pressures  of  the  pure  liquids  are  equal  are  shown  in  Fig.  75. 

167.     The  Gibbs-Konowalow  Rule. 

The  direction  of  change  of  pressure  occurring  in  the  distillation 
of  a  mixture  of  changing  composition  is  fixed  by  a  very  general 
rule,  deduced  by  Gibbs  (1876),  and  used  by  Konowalow  as  a 
consequence  of  some  experiments  of  Pliicker  (1854),  who  found 
that  the  vapour  pressure  over  a  mixture  of  alcohol  and  water  is 
all  the  less  the  larger  the  space  which  the  vapour  has  to  saturate. 
The  rule  may  be  stated  as  follows : — 

During  isothermal  increase  of  volume  of  a  vapour  standing  in 
stable  equilibrium  with  a  liquid  mixture,  the  total  vapour  pressure 
either  decreases  or  remains  unchanged. 

For,  if  we  suppose  that  a  portion  of  the  liquid  be  evaporated 
isothermally  and  reversibly  (say  by  raising  a  piston  enclosing  it 
and  the  vapour  in  a  cylinder),  any  resulting  change  of  composi- 
tion may  be  utilised  to  perform  work  by  remixing  vapour  and 
liquid  through  semipermeable  membranes.  Hence,  since  no 
work  must  be  done  on  the  whole  (§  163),  it  follows  that  a  com- 
pensating amount  of  work  has  been  spent  on  the  system  during 
the  evaporation,  and  hence  the  pressure  must  have  been 
diminishing  during  that  operation. 

Now  suppose  that,  at  a  particular  composition  of  the  liquid, 
the  total  pressure  increases  as  the  liquid  becomes  richer  in  a 
specified  component,  say  I.  Expansion  (i.e.,  distillation)  can  then, 
by  the  above  rule,  only  diminish  the  concentration  of  I.  in  the 
liquid.  The  concentration  of  I.  in  the  vapour  is  therefore,  at 
that  point,  not  less  than  its  concentration  in  the  liquid,  for  if  it 
could  be  less,  any  possible  evaporation  would  necessarily  increase 
the  concentration  of  I.  in  the  liquid. 

If,  however,  the  total  pressure  is  decreased  by  increasing  con- 
centration of  a  specified  component  in  the  liquid,  compression 
(i.e.,  liquefaction)  cannot  increase  the  concentration  of  that  com- 
ponent in  the  liquid,  and  hence  its  concentration  in  the  vapour  is 
not  greater  than  that  in  the  liquid. 

It  therefore  .follows  that  a  transition  from  a  rising  to  a  falling 
part  of  a  vapour  pressure  curve  can  occur  only  when  the  concen- 
tration of  a  specified  component  in  the  vapour  is  neither  greater 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    385 

nor  less  than  its  concentration  in  the  liquid,  or  in  other  words, 
it  can  occur  only  when  the  compositions  of  liquid  and  vapour  are 
identical.  Hence  we  deduce  the  exceedingly  important  Theorem 
of  Konoicaloic  :  A  liquid  mixture  corresponding  to  a  maximum  or 
minimum  of  rapour  pressure  at  any  specified  temperature  lias  the 
same  composition  as  the  rapour  in  equilibrium  irith  it. 

(a)  If  one  component  is  completely  non-volatile,  as  in  a  salt 
solution,    the   vapour  can  never  have  the    composition   of   the 
liquid,   and   the  curve  of    vapour  pressure  nowhere  exhibits  a 
transition  from  rising  to  falling,  i.e.,  it  possesses  no  maxima  or 
minima. 

(b)  The  converse  of  Konowalow's  theorem  states  that  if  there 
are  two  mixtures  of  slightly  different  composition,  one  of  which 
is  richer  in  a  specified  component  than  its  vapour,  and  the  other 
poorer,  there  is  an  intermediate  composition  at  which  the  composi- 
tions of  liquid  and  vapour  are  identical,  and  there  the  vapour 
pressure  has  a  stationary  value. 

If  the  evaporation  is  performed  under  isopiestic  conditions, 
instead  of  isothermal,  as  is  usually  the  case  in  practice  (/>  =  1 
atmosphere),  the  vapour  pressure  curves  must  be  replaced  by  the 
boiling  point  curves.  For  this  purpose  a  horizontal  line  corre- 
sponding to  the  given  pressure  is  drawn  across  the  vapour 
pressure  diagram.  The  abscissae  of  the  points  of  intersection  of 
this  line  with  the  various  isotherms  will  determine  the  composi- 
tions of  the  liquids  which  emit  vapour  of  the  specified  pressure 
at  the  temperatures  corresponding  to  the  various  isotherms  ;  and 
if  the  latter  are  plotted  against  the  compositions  one  obtains  the 
curve  of  boiling  points  under  the  given  pressure.  This  may 
be  repeated  for  other  pressures,  and  a  series  of  boiling  point 
curves  constructed.  The  reverse  (and  more  practical)  construc- 
tion is  similarly  effected.  In  particular,  it  is  easily  seen  that  points 
of  maximum  or  minimum  vapour  pressure  will  give  rise  to  points 
of  minimum  or  maximum  boiling  point  respectively,  with  the 
same  abscissae. 

The  gradients  of  the  partial  pressure  curves  at  the  abscissa 
corresponding  to  a  maximum  or  a  minimum  must  be  equal  but  of 
opposite  sign.  For  if  P  is  the  total  pressure, 


and 

dx 

1. 


386 


THERMODYNAMICS 


- 
' '  dx  ~        dx 

(J.  W.  Gibbs,  Trans.  Acad.  Conn.  (1875),  3,  155 ;  Sdentij. 
Papers,  I. ;  Konowalow,  Wied.  Ann.,  1881,  4,  48;  Zawidski, 
Zeitschr.  pliysik.  Chem,,  35,  129,  1900;  Meyerhoffer,  ibid.,  46, 
379,  1903). 

It  is  a  consequence  of  the  Gibbs-Konowalow  rule  that  the  com- 
positions of  liquid  and  vapour  (i.e.,  the  residue  and  distillate, 
respectively)  alter  in  the  sense  of  falling  and  rising  parts  of  the 
curve,  respectively.  One  may  imagine  (Ostwald,  loc.  cit.)  the 
composition  of  the  liquid  to  be  represented  by  a  heavy  mobile 

point,  that  of  the  vapour  by  a 
small  balloon,  constrained  to 
move  along  the  curve.  The 
former  tends  always  to  sink  to 
the  part  of  lowest  'pressure,  the 
latter  to  rise  to  the  part  of 
highest  pressure.  If  we  repre- 
sent the  different  types  of  curves 
in  one  diagram,  as  in  Fig.  76, 
and  draw  two  vertical  lines  a 
and  l>,  to  cut  all  the  curves,  it 
is  easy  to  see  what  will  be  the 
effect  of  isothermal  distillation 
on  the  mixtures  of  compositions 
a  and  b.  On  curves  (1),  (2),  (4)  the  composition  of  the  liquid 
must  move  to  the  left,  that  of  the  vapour  to  the  right.  The 
distillate  therefore  contains  a  greater  amount  of  the  more  volatile 
component  than  does  the  residue.  By  distilling  off  a  portion, 
repeating  the  process  with  the  distillate,  and  so  on,  at  the  same 
time  adding  the  residue  in  the  retort  to  the  former  residue,  we 
obtain  distillates  and  residues  containing  increasing  concentrations 
of  the  more  and  the  less  volatile  constituents,  respectively,  and 
the  separation  may  be  made  as  nearly  complete  as  we  please 
("  Fractional  Distillation").  On  the  curve  (8)  the  distillate  must 
be  at  the  commencement  the  mixture  of  maximum  vapour  pres- 
sure, for  otherwise  the  pressure  would  increase  during  distillation. 
The  two  components  are  therefore  withdrawn  from  the  liquid  in 
the  proportions  required  for  the  mixture  of  maximum  vapour 
pressure.  If  the  original  liquid  has  this  composition,  it  evapor- 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS   387 

ates  unchanged ;  if  not,  the  withdrawal  of  both  components  goes 
on  until  that  which  happens  to  have  been  in  excess  remains  in 
the  pure  state  in  the  retort.  In  the  case  of  a,  this  will  be  A,  in 
the  case  of  b,  it  will  be  B,  because  the  massive  mobile  point  will 
run  down  the  curve  to  A  or  B,  respectively,  whilst  the  light 
point  rises  to  M. 

If  the  curve  is  of  type  (5),  the  vapour  passing  off  will  be  pure 
A  or  B,  respectively,  according  as  the  mixture  is  a  or  />,  whilst 
the  residue  approaches  the  composition  M2.  When  this  composi- 
tion is  reached  the  whole  mass  distils  off  unchanged,  like  a  pure 
liquid. 

Mixtures  -of  this  type  have  been  known  for  a  long  time. 
Bineau  (1843)  found  that  the  solution  of  hydrochloric  acid  in 
water  which  distilled  off  unchanged  under  atmospheric  pressure 
had  approximately  the  composition  HC1.8H20.  The  similarly 
behaving  solution  of  nitric  acid  was  represented  as  2HN03.3H20. 
He  therefore  advanced  the  hypothesis,  which  has  been  obstinately 
maintained  by  some  chemists,  that  these  solutions  are  definite 
compounds.  Roscoe  and  Dittmar  (1860—61)  found,  however, 
that  if  the  pressure  is  altered,  the  composition  of  the  mixture  of 
maximum  boiling  point  is  altered  as  well,  so  that  the  locus  of  the 
maxima  on  the  boiling  point  curves  is  a  Hue  which  is  inclined  to 
the  T-axis.  Since  all  points  on  this  line  correspond  to  mixtures 
of  maximum  boiling  point  under  the  correlated  pressures,  and 
since  only  a  few  of  these  points  have  abscissae  corresponding  to 
simple  stoichiometric  proportions,  it  is  evident  that  we  must 
either  reject  Bineau's  hypothesis,  or  else  return  to  Berthollet's 
view  of  the  nature  of  chemical  compounds.  The  existence  of 
definite  compounds  in  the  solution  is  not,  of  course,  in  question 
here  at  all;  all  that  is  asserted  is  that  the  maximum  boiling 
solutions  are  not  necessarily  pure  compounds. 

The  theorem  of  Konowalow  is  the  basis  of  the  remarkably  interesting 
Faraday  Lecture  to  the  London  Chemical  Society  given  by  Ostwald  in  1904. 
He  points  out  that  the  considerations  which  have  been  summarised  above 
in  connection  with  Konowalow's  curves  lead  to  the  general  law  that  it  is 
possible  in  ever//  rase  to  separate  solutions  into  a  finite  number  of  hylotropic 
l>o<lies.  A  hylotropic  body  is  characterised  by  the  circumstance  that  it 
may  be  converted  completely  from  one  phase  into  another  in  a  given  process 
under  fixed  conditions  (e.y.,  constant  temperature  or  constant  pressure) 
without  variation  of  the  properties  of  the  residue  and  of  the  new 
phase.  The  apparent  exceptions  furnished  by  mixtures  of  maximum  or 

c  c  2 


388  THERMODYNAMICS 

miiiimum  vapour  pressure  vanish,  as  is  evident  from  the  results  of  Koscoe 
and  Dittmar,  if  the  distillation  is  carried  out  under  different  pressures.  The 
loci  of  maxima  or  minima  will  have  different  forms  according  to  the  nature 
of  the  solutions,  but  among  all  these  different  forms  there  is  one  distinguish- 
ing case,  viz.,  a  vertical  straight  line.  This  means  that  the  composition  is 
independent  of  pressure,  and  therefore  of  temperature,  and  when  this  is  the 
case  the  hylotrope  is  called  a  substance,  or  a  chemical  individual.  A  chemical 
individual  is  therefore  a  body  which  can  form  hylotropic  phases  within  a 
finite  range  of  temperature  and  pressure.  If  the  limits  of  this  range  be 
exceeded,  it  begins  to  behave  as  a  solution,  i.e.,  the  locus  bends  to  one  side 
or  the  other.  This  corresponds  to  dissociation  of  the  compound.  There  are, 
however,  some  substances  whose  range  of  existence  covers  all  accessible 
states.  These  are  the  elements . 

168.     The  Phase  Rule  of  Gibbs. 

Consider  a  system  of  a  number  of  phases  A,  B,  C,...R,  made 
up  of  the  components  a,  /3,  7,  .  .  .  v  in  different  proportions,  or 
of  the  same  substance  in  different  states  of  aggregation.  Let  the 
whole  system  be  free  from  the  action  of  gravity,  and  electrical 
or  magnetic  forces,  and  let  the  amounts  of  surface  energy  located 
on  the  various  interfaces  be  negligibly  small  in  comparison  with 
the  internal  energies  of  the  various  phases. 

Then  if  any  two  phases  are  separately  in  equilibrium  with  a 
third  phase,  they  are  also  in  equilibrium  when  placed  in  contact, 
so  that  if  any  one  phase  (e.g.,  the  vapour)  is  taken  as  a  test- 
phase,  and  the  other  phases  are  separately  in  equilibrium  with 
this,  the  whole  system  will  be  in  equilibrium.  Under  the  condi- 
tions imposed,  it  is  sufficient  that  the  vapour  pressure,  or  osmotic 
pressure,  of  each  component  has  the  same  value  at  all  the  inter- 
faces, for  we  may  consider  each  component  separately  by  intrud- 
ing across  the  interface  a  diaphragm  permeable  to  that  compo- 
nent alone.  Then  if  the  vapour,  or  osmotic  pressures,  are  not 
equal  at  the  third  interface  to  their  values  at  the  first  and  second 
interfaces,  i.e.,  at  the  interfaces  on  the  test-phase,  we  could  carry 
out  a  reversible  isothermal  cycle  in  which  any  quantity  of  a 
specified  component  is  taken  from  the  test-phase  to  the  phase  of 
higher  pressure,  then  across  the  interface  to  the  phase  of  lower 
pressure,  and  then  back  to  the  test-phase.  In  this  cycle,  work 
would  be  obtained,  which  however  is  impossible.  Hence  the  two 
phases  which  are  separately  in  equilibrium  with  the  test-phase 
are  also  in  equilibrium  with  each  other.  This  may  be  called  the 
Law  of  the  Mutual  Compatibility  of  Phases  (cf.  §  106). 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    389 

We  shall  now  assume  that  it  is  possible  to  have  a  system  in 
equilibrium  composed  of  the  various  phases  at  a  specified  tem- 
perature and  total  pressure.     This  will  be  characterised  by  certain 
definite  relations  between  the  compositions  of  the  phases  (for 
example,  a  solid  salt,  saturated  solution,  vapour  of  the  solvent). 
Let  TT,  T  =  total  pressure,  and  temperature,  of  the  system. 
n     =  number  of  components  (cf.  §  84). 
>•     =       „  phases  „ 

For  each  pair  of  phases  there  must  be  some  condition  (the 
exact  nature  of  which  is  immaterial)  satisfied  for  each  component, 
such  that  this  component  does  not  pass  from  one  phase  to  the 
other.  The  r  phases  may  now  be  arranged  in  (r  — 1)  pairs,  viz., 
(I,  2),  (2,  3),  (3,  4)  ...  {(/•  —1),  r},  and  there  will  be  therefore 
>•  —  1  conditions  to  be  satisfied  for  each  component  in  all  the 
phases,  that  is,  (r  —  1)  equations  defining  the  state  of  equilibrium. 
It  is  evident  that  such  pairs  as  (1,  3),  (2,  4),  ...  need  not  be  con- 
sidered, since  if  phase  (1)  is  in  equilibrium  with  phase  (2),  and 
phase  (2)  with  phase  (3),  then  (1)  will  also  be  in  equilibrium  with 
(3),  by  the  law  of  Mutual  Compatibility  of  Phases. 

For  all  the  components  there  will  be  /?(/•  —  1)  equations. 
The  number  of  variables  is  made  up  of : 
(i.)  the  pressure  IT, 
(ii.)  the  temperature  T, 

(iii.)  the  r(n  —  1)  independent  concentrations,  since  for  each 
phase  there  are  (n  —  1)  fractions  of  the  total  mass  for 
the  n  components,  the  last(n-th)  fraction,  being  obtained 
by  subtraction  of  the  sum  of  all  the  others  from  the 
total  mass,  is  fixed,  and  is  not  an  independent  variable. 
Hence  : 

Total  number  of  variables  =  2  +  ''("  —  !)• 
The  number  of  variables  left  undetermined 

=  number  of  variables  —  number  of  equations 
=  2  +  r(n  -1)  -  n(r  -  1) 
=  2  +  w  —  r. 

This,  however,  is  defined  (§  84)  as  the  number  of  degrees  of 
freedom  (F)  of  the  system,  hence : 

F  +  ;•=«  +  2      .         .         .         .     (a) 
which  is  Gibbs's  Phase  Rule.* 

*  Cf.  Partington,  Proc.  Chem.  Soc.,  191 1. 


390  THERMODYNAMICS 

If  any  component  is  absent  from  a  particular  phase  (for 
example,  the  vapour  phase,  or  a  solid  phase),  there  is  one  vari- 
able the  less,  but  also  one  boundary  condition  the  less,  for  migra- 
tion of  that  component  cannot  occur  with  respect  to  the  phase 
considered.  Equation  (a)  is  therefore  still  true. 

This  very  simple  rule  has  had  an  almost  incredible  influence  on  the 
development  of  theoretical  and  experimental  chemistry.  In  the  previously 
obscure  but  technically  very  important  department  of  the  study  of  mixed 
metals,  to  quote  a  single  example,  it  has  proved  invaluable. 

169.     Kirchhoff's   Equation. 

Previous  to  the  researches  of  Konowalow,  the  vapour  pressures 
of  mixtures  had  been  investigated  theoretically  by  G.  Kirchhoff 
(Pogg.  Ann.  (1858),  103,  104;  Ostw.  Khtss.  No.  101),  and  by 
Gibbs  (Scientific  Papers,  Vol.  I.).  The  latter  had  established  the 
theorem  relating  to  mixtures  with  stationary  vapour  pressures. 

If  two  liquids  are  mixed  together,  there  is  in  general  a 
change  of  intrinsic  energy  (AU)  and  a  change  of  free  energy  (A*). 
The  heat  absorbed  when  1  mol  of  [1]  and  x  mols  of  [2]  are  mixed 
in  a  calorimeter  is  the  increase  of  intrinsic  energy,  and  is  usually 
denoted  by  Q(ar)  : 

AU  =  Q(a)          ....     (1) 

Q(x)  may  be  positive  (heat  absorbed,  e.g.,  phenol  +  water),  or 
negative  (heat  ecolved,  e.g.,  sulphuric  acid  +  water)  ;  it  will  in 
general  be  a  function  of  x  and  of  temperature,  but  changes  only 
very  slightly  with  pressure. 

Thus,  with  water  and  sulphuric  acid,  Thorn  sen  found  at  the 
ordinary  temperature,  for  H2S04  +  ,rH20  : 


If  r,  r"  are  the   heat  capacities  of  the  unmixed  and  mixed 
systems  respectively  (§  58)  : 


If  F'  =  F,  the  heat  of  admixture,  Q(*'),  is  independent  of 
temperature. 

In  his  original  investigation,  Kirchhoff  proceeded  to  deduce  an 
equation  connecting  the  heat  of  admixture  of  two  liquids,  one  of 
which  is  very  sparingly  volatile,  with  the  vapour  pressure  of  the 
volatile  component  over  the  mixture. 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    391 

Let  us  consider  a  system  formed  of  a  homogeneous  mixture  of 
NI  mols  of  a  non-volatile  solute  with  N2  mols  of  a  volatile 
solvent,  together  with  a  further  >?2  mols  of  pure  liquid  solvent. 

The  system  has,  at  an  assigned  (arbitrary)  pressure  and  tem- 
perature (P,  T)  a  potential : 

</>  =  ,^2(P,  T)  +  $'(Xlf  N2,  P,  T)  .  .  (1) 
where  fa,  <£'  are  the  potentials  of  a  niol  of  the  pure  solvent,  and 
a  quantity  of  solution  such  that  NX  -f  N2  =1,  respectively. 

It'  V  is  the  total  volume  of  the  homogeneous  mixture  : 

V  =  (N1.+  Xa)f      ....     (2) 
where  T  may  be  called  its  mean  molecular  volume. 

Now  f>'(N%p2'  P?  T)  =  V  =  (N,  +  N2)f(;r,  P,  T)  .  (3) 
where  : 

.-Bjf*!-.- ^  W 

are  the  molecular  fractions,  or  numerical  concentrations,  of  the 
components. 

The  chemical  potential  /z-2,  of  the  solvent  in  the  solution  is,  by 
definition  : 


,,=  «*•'> 


_ 
f)N2  ""  aF 

c>t' 


'^.1 2  <-  **' 

This  relation  is  frequently  applicable. 

The  heat  absorbed  in  any  reversible  isothermal-isopiestic  change 
is  (§  55  (10) )  given  by  : 

dQ  =  e*($-T?*)    ....    (7) 

The  heat  absorbed  when  a  niol  of  solvent  is  added  reversibly  to 
the  solution  at  constant  temperature  (T)  and  pressure  (P)  is  the 
heat  of  dilution,  Q  (x,  P,  T). 


since  dtt.2  =  — 


392  THERMODYNAMICS 

The  heat  absorbed  when  a  mol  of  the  solvent  is  evaporated  at 
a  constant  temperature  T  from  a  volume  of  solution  so  large  that 
no  perceptible  change  of  concentration  occurs  during  the  process, 
is  called  the  heat  of  volatilisation  \  (x,  T).  From  (7)  : 


A  (r  rm  _         i,   *  i     _        T    _ 

-8N2~~ 

where  p  is  the  pressure  corresponding  to  a  concentration  x, 
and  92  is  the  potential  of  a  mol  of  the  vapour  of  the  solvent 
under  the  given  conditions. 

The  heat  of  volatilisation  of  the  pure  solvent  under  its 
saturated  vapour  pressure,  ir,  at  the  temperature  T  is 
similarly  : 


A(T)  =     fc  (»,T)  -  *(,,T)     -  T  -  .  (10) 


Then  E,  —  >     >       -  T  —    rd-P   T)  —  T 

1-  A~n'    '  8T 


.     .     .  (11) 

We  shall  now  assume  that  r  is  negligibly  small,  then  EI  is 

independent  of  the  pressure  P      ..       ..       .        •         •         •     (a) 

If  «2  (P,  T)  is  the  molecular  volume  of  the  pure  liquid  solvent : 


(P,  T)  -  T  =  »,  (P,  T)  -  T     «  =  E2,  say. 


If   «2   is   negligibly   small,   E2   is   independent   of  the    pres- 
sure P  .         .        ......         .        .         .     (b) 


Further,  1  (92  (p,  T)  -  T  8?2^'  T^)  =  V2  (p,  T) 


where  ¥2  is  the  molecular  volume  of  the  vapour. 

But  if  the  vapour  obeys  the  gas  laws,  E3  =  0        ,     (c) 
Thence  we  can  write  : 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    393 


as  we  see  from  the  expressions  for  EI,  E2,  E3  and  the  condition 
that  these  are  all  independent  of  the  pressure. 
From  (a),  08),  (7)  and  (8),  (9),  (10)  it  follows  that : 

Q  (.r,  T)  =  A  (x,  T)  -  A(T)    .         .         .     (13) 

This  equation,  it  must  be  observed,  is  not  true  generally,  but 
only  on  the  assumptions  introduced  : 

(1)  r,  «2  are  negligible ; 

(2)  The  saturated  vapour  obeys  the  gas  laws. 
We  shall  now  find  an  expression  for  A  («r,  T). 

Let  the  system  consist  of  the  solution  as  before,  in  contact 
with  »2  mols  of  the  vapour  of  the  solvent,  under  a  pressure  j>. 

The  increase  of  entropy  when  SN-2  niols  of  solvent  are  removed 
from  the  solution  in  the  form  of  vapour  at  the  constant  tempera- 
ture T  and  pressure  p  is  : 

£cj  A  \tl'9   -L  )  £v" 

-f^ 

But     S  =  -£*  =  -  ~ 


_    _ 

L        f>T  ?T 

,  T)  =    -  T 


We  shall  next  examine  the  quantity  enclosed  in  the  brackets. 
The  solution  is  in  equilibrium  with  its  vapour  when  the  condensa- 
tion of  an  infinitesimal  amount  of  the  latter  leaves  the  whole 
potential  of  the  system  unchanged,  i.e.,  the  changes  of  potential 
of  solution  and  vapour  are  equal  and  opposite  ; 


394  THERMODYNAMICS 


.  (15) 

i.e.,  the  chemical  potentials  of  solvent  in  solution   and  vapour  are 
equal. 

If  Ma  ^  92,  evaporation,   or  condensation,   respectively,    will 
occur. 

Put  ft  (x,  p,  T)  -  ?2  (p,  T)  =  F  (x,  p,  T) 

d¥(x,p,T)      a/i  (./-,_/>,  T) 
tnen  ^ =  — —      J 

dp 


ST 


Thence  X  (j?,  T)  =  T    v2  +  x       -  p       .         .         .  (16) 


This  is  a  very  important  general  equation. 
Again  we  assume  that  r  is  negligible  in  comparison  with  V2, 
and  that  the  saturated  vapour  obeys  the  gas  laws. 

/.  A  (.r,  T)  =  RT*  1  e>^l>  =  BT^fa^>T)         §  (ly) 

which  is  analogous  to  the  Clapeyron-Clausius  equation  and  is  due 
to  L.  Natanson  (1892). 


We  have  also  A(T)  =  BT2  .         .   -     .         .     (18) 

for  the  pure  liquid  solvent  (§  88). 

Thus,  from  (13),  (17),  and  (18)  we  have  : 

^       '..         .     (19) 

This  equation  was  deduced  by  G.  Kirchfaqff(P<w$r.  Ann.,  1858.  Cf. 
Jiittner,  Zeitschr.  i)h)tsik.  Chem.,  38,  76,  1901  ;  Scholz,  Wied. 
Ann.,  45,  193,  1892  ;  Schiller,  ibid.,  67,  292,  1899.) 

Since  Q  (x,  T)  differs  from  Q  (x,  T),  the  calorimetric  heat  of 
admixture,  only  by  the  small  amount  of  external  work,  we  may 
in  all  cases  where  liquids  or  solids  only  are  concerned  use  the 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    395 

latter  thermal  magnitude  iu  (19).  Thus  Kirchhoff,  from 
Thomson's  results  for  the  heat  of  admixture  of  water  and  sul- 
phuric acid,  calculated  the  vapour  pressure  curve  of  mixtures  of 
these  two  substances,  and  obtained  values  in  fair  agreement  with 
the  numbers  of  Regnault.  The  converse  problem  is  more  uncer- 
tain, on  account  of  the  large  influence  of  small  errors  in  the 
determination  of  vapour  pressure. 

The  sign  of  Q  (x,  T)  determines  the  vapour  pressure  law  of  the 
solution : 

(1)  If  Q  (x,  T)>  0,  i.e.,  heat  is  absorbed  on  dilution  (as  is  the 

case  with  a  large  number  of  salt  solutions)  then  In  —  ,  and  hence 

the  relative  lowering  of  vapour  pressure  -  — — ,  decreases  with 

rise  of  temperature. 

(2)  If  Q  (x,  T)  <  0,  i.e.,  heat  is  erolretl  on  dilution  (H2S04. 
CaCla,  NaOH).  the  relative  lowering  increases  with  rise  of  tem- 
perature. 

(3)  If  Q  (x,  T)  =  0,  i.e.,  the  heat  of  dilution  is  zero,  the  rela- 
tive lowering  is  independent  of  temperature.     Thus,  the  necessary 
and  sufficient  condition  for  the  validity  of  the  law  of  von  Babo 
(§  130)  is  that  the  heat  of  dilution  is  zero  at  all  temperatures  and 
concentrations  in  the  range  considered. 

In  the  above  investigation  Q  (,r,  T)  has  the  significance  of  a 
heat  of  dilution,  i.e.,  it  denotes  the  heat  absorbed  when  more 
solvent  is  added  to  a  solution  of  concentration  ,r.  If  we  consider 
a  solid  salt  which  is  dissolved  in  a  solvent,  Q  (,r,  T)  has  the  signi- 
ficance of  a  heat  of  solution.  If  we  consider  a  saturated  solution 
we  recover  the  case  treated  in  §  132.  The  vapour  pressure  p  of 
the  solution  is  now  a  function  of  temperature  alone,  since  the 
concentration  of  a  saturated  solution  is,  at  a  given  pressure, 
completely  defined  by  the  temperature,  and  it  alters  only  very 
slightly  with  change  of  pressure. 

170.     The  Duhem-Margules  Equation. 

The  equation  of  Kirchhoff  applies  to  the  case  of  a  volatile 
solvent  and  an  involatile  solute ;  we  shall  now  consider  the  case 
of  a  mixture  of  two  volatile  substances,  i.e.,  a  binary  mixture  in 
the  sense  of  §  163.  We  use  the  following  notation  : 


396  THERMODYNAMICS 

NI,  N2  =  numbers  of  mols  of  the  two  components,  [1]  and  [2], 

in  the  liquid  phase  ; 

nlt  «2  =  numbers  of  mols  in  the  vapour  phase  ; 
Plf  P2  =  vapour   pressures   of   pure    [1]  and   [2]   at  a  given 

temperature  T  ; 
plt  p2  =  partial  pressures  of  [1]  and  [2]  in  the  vapour  of  the 

mixture  at  the  temperature  T. 

The  molecular  fractions,  or  the  numerical  concentrations  in  the 
liquid,  are: 

Ni       .    (l        .  _ 
--'  - 


N2 


We  shall  now  calculate  the  diminution  of  free  energy  which 
results  from  the  admixture  of  NI  mols  of  [1]  and  N2  mols  of  [2], 
both  in  the  liquid  state.  The  simplest  method  is  an  application 
of  equation  (13)  of  §  52,  which  states  that  the  work  done  in  the 
isothermal  and  reversible  execution  of  a  process  is  equal  to  the 
diminution  of  free  energy  : 

SAT  =  —  tl  * 
.-.    AT=  -  f  rf*  =  *0  —  * 

where  ^o,  *  refer  to  the  initial  and  final  states.  It  follows  from 
Moutier's  theorem  (§  36  ;  cf.  §  58)  that  SPo  —  *  is  equal  to  minus 
the  least  possible  amount  of  work  which  must  be  spent  in 
separating  the  mixture  into  the  two  liquid  components,  by  any 
isothermal  reversible  process.  We  shall  select  the  process  of 
isothermal  distillation,  introduced  by  Kirchhoff  (1858).  Let  each 
component  be  removed  in  the  state  of  vapour  through  two  semi- 
permeable  partitions,  so  that  the  operation  proceeds  isothermally 
and  reversibly,  and  with  unchanging  composition  of  the  liquid. 
Each  component  is  therefore  removed  under  a  constant  pressure, 
equal  to  its  partial  pressure  over  the  mixture. 

It  is  further  assumed  that  : 

(i.)  Both  vapours  obey  the  gas  laws  ; 

(ii.)  The  volume  of  the  liquid  is  small  in  comparison  with  that 
of  its  vapour. 

The  operations  are  as  follows  : 

(1)  Evaporate  out  NI  mols  of  (1)  and  N2  mols  of  (2).  The 
amount  of  work  done  is  : 

m  +  p&9  =  (Nj  +  N2)  BT. 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    39? 
(2)  Compress  the  vapours  to  PI,  P2.     The  work  done  is  : 


-RT     Ni  Zn£i  +  N2  ln^   . 
V    '        pi  pi> 

(3)  Condense  on  the  liquids.     The  work  done  is : 

-  (Pi  Vi  +  P2Va)  =  -  (Nx  +  N2)  RT. 
The  total  expenditure  of  work  =  loss  of  free  energy 


*. 

To  obtain  the  required  relation  between  the  partial  pressures 
and  the  concentrations  in  the  liquid,  we  suppose  a  very  small 
quantity  SNi  mols  of  (1)  is  distilled  isothermally  and  reversibly 
from  the  pure  liquid  (1)  to  the  mixture  of  NX  mols  (1)  +  N2 

mols  (2).     The  work  done  is  : 

p 

—  S  [A  *]  E  —  S*  =  SNi.  RT  In  -1  (Na  const.) 

Pi 

.'.  in  the  limit,  with  unaltered  concentrations  : 

7}  &  P 

—  ^  =  RT  In  —  (N2  const.) 
dJNi  _/>i 

Similarly  -  |^  =  RT  In  ^  (Ni  const.) 

But  Ni  =  N2 


l  —  x 

...       d  Ni  =  N2      ^'     2  (N2  const.) 

...  _iy?  =  B?y?»  !»£  ....  (2) 

Differentiate  (1)  with  respect  to  a-,  and  compare  with  (2) : 


_.rnvr 

~-  R1Na 


i         Inpi         djnpfi 

~  (i-*f~  fa  J 


(3) 


Thence  +(l_*)  =  0     .         .         .     (4) 


398  THERMODYNAMICS 

Equation  (4)  is  the  fundamental  differential  equation  in  the 
theory  of  binary  mixtures;  it  was  obtained  by  Duhem  in  1887 
( Traite  de  Mecaii.  Chim.  IV.). 

Duhem's  equation  was  integrated  by  Margules  (Sitzungsber. 
Wien  Akad.,  1895)  by  means  of  the  substitutions : 


The  a's  aud/3's  are  constants  fora  fixed  temperature  and  given 
components. 

If  one  curve,  say  pi  =  </>(.*•),  is  known,  the  other,  and  hence 
the  total  pressure  curve,  is  known  from  (4)  for  all  concentrations, 
and  conversely  if  the  total  pressure  curve,  II  =  F  (x),  is  known, 
both  partial  pressure  curves  are  determined  when  (4)  is  written  in 
the  form  : 

(xfl  —  7^)  -^  +  (1  —  x)p,  TT-  (6) 

ox  *    dx 

The  coefficients  of  (5)  are  related  as  follows : 

/JO  —  #o  —  ^1  >    Pi  —~   —  #1  j    Pa  — *  #2  ~\~  &3  ~T"   •  •  • )  r» 

&=  -a3-2a4-aa5-...  \    '     (/) 

The  equations  of  the  tangents  to  the  partial  pressure  curves 
are  : 


—  ',  /  O  \ 

~  ' 


At  their  intersections  with  the  pressure  axes,  x  =  0  and  x  =  1  : 


f]M        _  I      *        *        '     ^9) 

But  if  we  assume  Raoult's  law  for  these  limiting  dilutions  : 

.'.  ao  =  ^o  =  1 ;  a\  =  fti  =  0     .         .         .   (11) 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    399 
From  (5)  and  (11)  : 


Differentiate  (12)  with  respect    to    a?,  and   put   ,r  =  0,   then 
a;  =  1  : 


/i'M  ^f- 

\?7/.r=i~~  \i 

which  are  the  equations  of  the  tangents  of  the  angles  of  contact, 
and  are  analytical  expressions  of  Henry's  law,  as  we  see  on 
putting : 


=  const.  =  a 


e  ~  =  const.  =  I 

where  a,  b  have  the  significance  of  solubility  coefficients  tZawidski, 
Zeitschr.  physik.  Chem.,  69,  1909  ;  cf.  Story,  ibid.,  71,  129,  1910  ; 
Rosanoff  and  Easley,  ibid.,  641  ;  R.  Plank,  Phys.  Zeitschr.,  11,, 
49,  1910;  Konowalow,  Journ.  de  Phys.,  [iv],  7,  207,  1907). 

Let  S\,  Si'  be  the  volumetric  conceatratious  of  pure  [1]  iu  liquid  and 
vapour,  3-2',  S2"  similar  vahies  for  pure  [2].  Also  let  &',  |i",  and  &,  |2",  be 
the  concentrations  in  the  liquid  and  vapour  phases  of  the  solution. 

In  general  £>£,.„,  £>|; 
and  if 


then  <'."4<'- 

But  li"/fi'  =  «»  and  &",'&'  =  '' 


These  relative  solubih'ty  coefficients  completely  define  the  character  of  the 
partial  and  total  pressure  curves  of  binary  mixtures, 


400  THEBMODYNAMICS 

(i.)  If  all  the  as  and  /3's  are  zero  : 

a  =  b  =  1 
.  fcl'^fc" 
'  '  fi'  ft' 

i.e.,  the  concentrations  in  both  phases  are    equal  to   those   of   the    pure 
components  : 

li"       Si"          fe"       S2" 
TT  =  =rrand  -r-r  =  ,^-r 

ll  •*!  1-2  S2 

In  this  case  equation  (12)  leads  to  the  relations 
Pi  =  Pi* 

J>g  =  Pa(l    -.X) 

so   that   both   partial   pressure  curves,  and  the  total  pressure  curve,  are 
neutral  curves. 


where  k  is  a  constant  : 

*  =  Pi  /P8 
This  equation  is  similar  to  that  of  F.  D.  Brown  (1879), 

PI  p2  =  k  fir7> 

where  .»•'  =  concentration  in   the  vapour.     It  holds  good  for  mixtures  of 
benzene  with  ethylene  dichloride. 

For  other  non  -associated  and  closely  related  liquid  pairs,  equation  (lo) 
applies,  but  A-  is  not  usually  equal  to  Pi/Pa  and  has  to  be  determined  by 
experiment. 

(iL)If  3+1+  •  •    >0,and|  +  |!+.  ..    >0, 

a  >  l.and  6  >  1, 

$>%"*&>$• 

Both  partial  pressure  curves 

Pi  =/i(-'').  Pi  =./i  (•'•), 
lie  over  the  lines  joining  the  points 

(  2h  =  0  for  x  =  0 
(^1  =  ?!  for,r=l, 
I  ^  =  P2  for  ao  =  0 
(/*2  =  0  forx=-r. 
They,  and  the  corresponding  total  pressure  curve,  are  positive. 

.(iii)If  ?+!+   ..-    <0,and^+f  +.   .    <0 

a  <  1,  and  6  <  1, 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS  401 

£<£•-£<£. 

so  that  both  partial  pressure  curves,  and  the  total  pressure  curve,  are  negative. 

(iv.)  If  f-+y  +  •  •  >  0,  butJ+^+  -  -   <  0 

a>l,  but&<i, 
then  j£>IV,andfV<f£, 

?1  •=•!  «1  A* 

so  that  one  partial  pressure  curve  is  positive,  and  the  other  negative,  the 
total  pressure  curve  exhibiting  a  point  of  inflexion.  Curves  of  this  type  are 
said-  to  be  shown  by  mixtures  of  pyridine  with  water,  and  of  methyl  alcohol 
with  water. 

If  we  differentiate  (1)  with  respect  to  T  we  have  : 

_»W*)  =  B  [x.ta^  +  S.htO  +BT4 
L        lh  piJ  rT 

...  T  ?»*>  =  -  A*  +  VP  »   [31,6,  ?!  +  NV^ 
9T  rT  L       PI  j>2 

If  AU  =  Q  (Mi,Na,T)  is  the  heat  of  admixture  at  T.  we  have 
generally  (§  58)  : 


=  -  RT2       [NX/W    L+  NV"    ?     .        .         .   (16) 


Now  suppose  N-2  mols  of  pure  liquid  [2]  are  isothermally  and 
reversibly  distilled  into  NI  mols  of  pure  liquid  [!"•.  The  change 
of  free  energy  for  distillation  of  SN2  mols  of  [2]  into  a  mixture 
over  which  the  partial  pressure  is  p%  is,  as  we  have  shown : 

.*.  for  the  complete  process,  in  which  the  partial  pressure  p 
of  [2]  increases  from  0  to  p*t  and  its  concentration  from  0  to  .r,  the 
change  of  free  energy  is  : 

f*  P 

—  A  *  =  RT     /it  £-?  rfNa    (Ni  const.)  , 

Jo    ' 
Similarly,  for  distillation  of  [1]  into  [2] :  .    (18) 

-  A*  =  RT  !  In  ?i  dNi   (N2  const.) 


D    D 


402  THERMODYNAMICS 

/•Na 
...  RT     IH  1?  dNa  =  RT 


/• 

J  IH  1?  dNa  = 


Pi  Pz  n  p       2 

r  o 

These  relations  were  deduced  by  Nernst  (1893). 

171.     Dolezalek 's  Theory  of  Binary  Mixtures. 

The  vapour-pressure  curves  of  binary  liquid  mixtures  have  been 
considered  from  another  point  of  view  by  Dolezalek  (Zeitscher. 
physik.  Chem.  64,  727,  (1908)),  who  starts  out  with  the  very  simple 
assumption  that  the  partial  pressure  of  each  component  is  pro- 
portional to  its  concentration  in  the  liquid  phase,  provided  no 
chemical  change  occurs  when  the  liquids  are  mixed,  and  that  neither 
component  is  polymerised  in  the  liquid  state.  Thus  : 

—  P    v        ^A  _  p    v       N] 


The  neutral  curves  are  therefore  characteristic  of  non-associated 
substances,  a  conjecture  which  is  in  accord  with  the  results  of 
independent  branches  of  investigation. 

A  partial  pressure  curve  which  is  concave  to  the  concentration 
axis,  i.e.,  a  positive  curve,  indicates  the  dissociation  of  a  poly- 
merised component,  whilst  a  curve  which  is  convex  to  the  same 
axis,  i.e.,  a  negative  curve,  indicates  the  formation  of  a  chemical 
compound  of  the  two  components.  In  the  first  case  the  con- 
centration of  the  constituent  passing  into  the  vapour  would  be 
increased,  in  the  second  case  reduced,  by  the  assumed  change. 
As  examples,  Dolezalek  quotes  : 

(1)  No  association  or  combination — neutral  curves  : 

benzene  +  chloroform. 

(2)  Association — positive  curves : 

carbon  tetrachloride  +  benzene 
(CC14)2^  2CC14  (15  per  cent,  double  molecules  at  50°). 

(3)  Combination — negative  curves : 

chloroform  -f-  acetone 
CHC13  +  (CH3)2CO^:CHC13(CH3)2 .  CO. 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS  403 

The  proportion  of  polyniersed  substance,  or  of  the  compound, 
may  be  calculated  by  the  law  of  mass  action. 

Emil  Bose  (1910)  maintains  that  Zawidski's  calculations,  with  Margules' 
solution  with  only  a  few  coefficients,  are  not  satisfactory,  and  proposes  to  find 
the  partial  pressures  by  a  graphical  method  which  consists  in  drawing  the  two 
partial  pressure  curves  so  that  the  sum  of  their  ordinates  is  everywhere  equal 
to  the  ordinate  of  the  (known)  total  pressure  curves.  The  Duhem  equation 
shows  that  />!,#>  are  positive,  continuous,  and  single-  valued  functions  of  x, 
so  that  only  one  decomposition  of  the  total  pressure  curve  has  any  physical 
significance,  and  for  every  value  of  .»•  : 

.*•  _      <li>!  'fa 

~  i^I7:'      TT^r  • 

172.     Dolezalek's  Rule. 

Dolezalek  had  previously  (1903)  proposed  a  very  simple  relation 
between  the  vapour  pressures  of  concentrated  salt  solutions  and 
their  composition  ;  the  logarithm  of  the  vapour  pressure  of  the 
solvent  is  nearly  a  linear  function  of  the  number  of  mols  of 
salt  (x)  per  rnol  of  water  : 

—  Inp  —  ax  +  b          ......     (1) 


/.  —  _      —  a  —  a  characteristic  constant    .         .  (la) 
dx 

Now  according  to  Nernat  and  Roloff  (§  130)  : 

(l^P  +  x(!^  =  Q  (2) 

dx  dx 

where  p,P  are  the   vapour  pressures  of  the  solvent  and  of  a 
volatile  solute,  respectively.     Thence  : 

dlnP 

x  —  i  —  =  a 
dx 

or  /«P  =  alnx  -f-  const.  .         .         .     (3) 

so  that  the  logarithm  of  the  partial  pressure  of  the  solute  is  also 
a  linear  function  of  the  number  of  mols  per  rnol  of  water. 

Equation  (3)  was  verified   with  aqueous  solutions  of  hydro- 
chloric acid. 

173.     Thermal  Relations  of  Binary  Liquid  Mixtures. 

We  have,  so  far,  in  considering  the  evaporation  of  binary 
mixtures,  taken  account  only  of  the  partial  pressure  relations. 

D  D  2 


404  THE  RMOD  YN  AM  1C  S 

These,  however,  are  intimately  related  to  the  quantities  of  heat 
absorbed  in  the  formation  of  the  mixture,  and  in  its  evaporation. 
In  considering  the  thermal  magnitudes  of  importance  in  the 
study  of  binary  liquid  mixtures,  we  shall  confine  our  attention  to 
the  simplest  case,  in  which  it  can  be  assumed  that  : 

(1)  The  volume  of  the  liquid  is  negligibly  small  in  comparison 
with  that  of  its  vapour. 

(2)  The  vapour  is  a  mixture  of  ideal  gases  (cf.  §  122). 

From  a  very  large  volume  of  the  mixture  of  NI  mols  of  [1]  and 
N2  mols  of  [2],  let  a  very  small  amount  of  vapour  be  taken.  The 
concentration  may  be  assumed  to  be  constant  during  the  process, 
and  hence  also  the  total  pressure  : 

n=pi  +  p*  .     (1) 

If  iii,  "2  are  the  numbers  of  mols  in  the  vapour  : 

j>i  =  —  ^-n;  P,  =  _^_n      .       .    (2) 

MI  -j-  M2  "i  +  "a 

Let  Alt  A2  be  the  quantities  of  heat  absorbed  when  a  mol  of 
each  component  is  evaporated,  under  its  own  partial  pressure, 
from  a  large  volume  of  liquid  into  a  large  volume  of  vapour.  Then, 
from  hypothesis  (1)  : 

...     (3) 


We  put  -^  -  A!  +  _-!?*_  Aa  =  A.  ,     (5) 

Ml  +  «2  MI  +  M2 

the  heat  of  evaporation  of  a  mol  of  the  mixture.     Then  : 

/am     AH 

'         '         '         '         •    ' 


which     is    formally     identical     with     the    Clapeyron-Clausius 
equation. 

If  AI',  A2'  are  the  molecular  latent  heats  of  evaporation  of  the 
pure  liquids  at  the  same  temperature,  then  from  hypothesis  (2)  : 
A  =  NiAj'  +  N2A2'  -  W         .         .         .     (7) 

where  W  is  the  heat  absorbed  in  mixing  NI  mols  of  the  first 
liquid  with  N2  mols  of  the  second.    The  amounts  of  heat  absorbed 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS  405 

on  adding  a  niol  of  each  liquid  component  to  the  mixture  are 
%,  oW/3N2,  respectively.     Thus  : 
/?W\ 

S    '     •     •     •« 

\9N2/  T  J 
But,  if  PI,  P2  are  the  vapour  pressures  of  the  pure  liquids : 


(9) 


in   which   only   the  vapour   pressures   of   the    unmixed   liquids 
appear. 

Also,  from  (1)  and  (5)  : 
Pi  _       "i         _  A.  —  A2  _ 


II          »l  +  »2          ^1  —  ^2  '       11          "l  +  "2          Al  — 


Corollary.  —  The  change  of  composition  of  the  vapour  with 
temperature  is  determined  in  magnitude  and  sign  by  the 
difference  of  the  heats  of  evaporation  from  the  mixture. 

From  (3)  and  (10)  we  find  : 


...     (13) 

ar  \Pzl=  K '  RT*  '  VaNai 

i.e.,  the  partial  pressures  may  be  calculated  if  we  know  the 
pressures  of  the  unmixed  components,  and  the  heat  of  admixture 
as  a  function  of  temperature  and  of  the  composition  of  the 
mixture  : 

W  =  W(N1,N2,T) 

(R.  Luther,  1898 ;  P.  Duhem,  1899.) 


406  THERMODYNAMICS 

The  heat  of  admixture  of  a  mol  of  solution,  w,  is  defined  by  the 
equation  : 

W=(N  +  N2)w    .        ,      '  ."    •    .     (14) 

and  since  N^Ni  +  N2)  =  ./•    .         .         .         .     (15) 


Wehave 

-•:/'..:' 


_  _ 

^       xfa 


From  (13)  and  (16)  we  find  : 

«  +  V.-*)@y\ 

\ox  /  T 


so  that  we  have  the  equations  : 

w  =  RT2          a:Zn     -+  (1  -  *)  /»•          .         .     (17) 


which  are  due  to  Nernst  (1893).     (Of.  Duhem,  Traite,  IV.,  215 
et  seq.) 

Equation  (18)  was  applied  by  Margules  (1895)  to  calculate  the 
heats  of  admixture  of  water  and  alcohol  from  the  vapour-pressure 
data  of  Regnault  ;  the  results  agreed  with  the  direct  deter- 
minations of  Winkelmann  (1873). 

174.     Partially  Miscible  Liquids. 

There  are  two  very  important  generalisations  which  may  be 
established  in  connexion  with  the  second  type  of  liquid  mixtures 
investigated  by  Regnault,  e.g.,  mixtures  of  ether  and  water. 

(1)  So  long  as  tivo  layers  are  present  in  contact  with  vapour,  the 
composition  of  each  is,  at  a  specified  temperature  and  pressure, 
independent  of  the  absolute  or  relative  amounts  of  the  layers. 
Further  addition  of  either  component  therefore  leads  to  the  growth  of 
one  phase  at  the  expense  of  the  other,  but  the  concentration  of  each 
remains  unchanged. 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS  407 


This  is  a  consequence  of  the  phase  rule ;  there  are  two  com- 
ponents in  three  phases,  hence  the  number  of  degrees  of  freedom 
is  2,  so  that  when  the  temperature  and  pressure  are  fixed,  the 
composition  of  each  layer  is  also  defined. 

(2)  If  two  liquid  solutions  are  in  equilibrium  with  each  other, 
their  vapour  pressures,  and  the  partial  pressures  of  the  components 
in  the  vapour,  are  equal. 

Konowalow  established  this  important  rule  by  means  of  the 
following  reasoning.  The  two  liquid  layers  a  and  /3  are  con- 
tained in  a  ring-shaped  tube,  and  above  them  is  the  vapour.  The 
liquids  are  in  equilibrium  across  the  interface  A.  Then  if  the 
pressure  of  either  component  in  the  vapour  were  greater  over  a 
than  over  (3,  diffusion  of  vapour  would  cause  that  part  lying  over 
ft  to  have  a  higher  partial  pressure 
of  the  given  component  than  is 
compatible  with  equilibrium.  Con- 
densation occurs  and  ft  is  enriched 
in  the  specified  component.  By 
reason  of  the  changed  composition 
of  /?,  however,  the  equilibrium 
across  the  interface  is  disturbed, 
and  the  component  deposited  by  the 
vapour  ivill  pass  into  the  liquid  a. 
The  whole  process  now  commences 
anew,  and  the  result  is  a  never- 
ending  circulation  of  matter  round 

the  tube,  i.e.,  a  perpetual  motion,  ivhich  is  impossible.  Hence 
the  partial  pressures  of  both  components  are  equal  over  a 
and  /3,  and  therefore  also  is  their  sum,  i.e.,  the  total  vapour 
pressure. 

This  is  evidently  a  special  case  of  the  Law  of  Mutual  Com- 
patability  of  Phases  (§  168).  For  the  application  of  the  theory 
of  chemical  potential  to  such  systems  cf.  §  155. 

If  the  temperature  is  changed  the  miscibility  of  the  liquids 
alters,  and  at  a  particular  temperature  the  miscibility  may 
become  total;  this  is  called  the  critical  solution  temperature.  With 
rise  of  temperature  the  surface  of  separation  between  the  liquid 
and  vapour  phases  also  vanishes  at  a  definite  temperature,  and 
we  have  the  phenomenon  of  a  critical  point  in  the  ordinary  sense. 
According  to  Pawlewski  (1883)  the  critical  temperature  3  of  the 


408 


THERMODYNAMICS 


mixture  may  be  calculated  from  those  of  its  components 
by  the  mixture  rule  (§  120) : 

+ 


This  equation  is  only  approximate  (Kuenen,  1895),  but  usually 
gives  very  good  results  (Schmidt,  1891).  In  some  cases,  how- 
ever the  critical  point  may  be  even  lower  than  that  of  either 
component. 

175.     Vapour-Pressure  Curves  of  Partially  Miscible  Liquids. 

The  vapour-pressure  curves  for  such  mixtures  as  ether  and 
water  consist  of  three  parts  : 

(1)  and  (2)  The  end  curves  corresponding  with  the  homo- 
geneous solutions  of  B  in  A,  and  of  A  in  B. 

(3)  The  central  curve,  corresponding  with  the  two  layers. 
There  are  three  types  formally  possible,  viz.,  those  in  which 
the  end  curves  both  ascend  or 
descend,  or  one  ascends  and  the 
other  descends,  to  the  central 
curve.  Konowalow  found,  how- 
ever, that  the  second  type  does 
not  actually  occur,  so  that  the  two 
known  forms  are  a  and  /3.  The 
non-existence  of  the  other  type  is 
also  evident  from  the  consideration 
that  the  pressure  of  the  mean  curve 
lies  above  those  of  the  components 
in  a,  between  them  in  /?,  and  would 
lie  Icncath  them  in  the  other  case. 
Now  on  a  rising  part  of  the  curve 
the  concentration  of  the  compo- 
nent on  the  other  side  of  the  central  curve  must  be  greater  in  the 
vapour  than  in  the  liquid  phase,  for  a  falling  part  the  reverse  is 
true.  For,  taking  the  curves  starting  from  the  A  axis,  we  see 
that  addition  of  B  increases  the  pressure.  But  we  have  already 
proved  that  compression,  i.e.,  condensation  of  vapour,  always 
increases  the  pressure.  Hence  condensation  of  vapour  increases 
the  concentration  of  B  in  the  liquid,  hence  the  vapour  is  richer 
in  B  than  the  liquid.  If  the  curve  descends,  as  in  the  case  of  the 


100% 

A 


100% 
B 


FIG.  78. 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS  409 

part  of  a  coming  from  the  B-axis,  we  see  that  addition  of  A  raises 
the  pressure.  But  compression,  i.e.,  condensation  of  vapour, 
increases  the  pressure,  hence  the  vapour  is  richer  in  A  than  the 
liquid,  and  hence  poorer  in  B.  If  now  we  have  two  end  curves 
descending  to  a  mean  curve,  in  the  point  where  the  first  cuts  the 
mean  curve  the  vapour  is  poorer  in  B  than  the  liquid.  The  second 
point  of  intersection,  on  the  other  hand,  is  on  an  ascending 
portion  of  the  curve,  and  hence  the  vapour  is  richer  in  B  than  the 
liquid.  At  hoth  points,  however,  the  vapour  has  the  same  com- 
position (because  two  layers  are  present),  and  in  the  second 
point  the  liquid  is  much  more  concentrated  than  in  the  first. 
Thus  the  vapour  is  more  dilute  than  the  most  dilute,  and  more 
concentrated  than  the  most  concentrated  liquid,  which  is 
impossible.  One  end  curve  must  therefore  lie  below  the  central 
curve. 

176.     Immiscible  Liquids. 

If  two  liquids,  such  as  benzene  and  water,  do  not  mix  appre- 
ciably, the  vapour-pressure  relations  are  very  simple,  for  each 
constituent  emits  vapour  under  the  same  pressure  as  if  it  alone 
were  present.  The  total  pressure  is  the  sum  of  the  vapour 
pressures  of  the  pure  liquids  : 


If  the  temperature  is  raised,  both  partial  pressures  increase, 
and  when  their  sum  reaches  the  value  of  the  total  pressure  to 
which  the  free  surface  of  the  mixture  is  exposed  (usually  the 
pressure  of  the  atmosphere)  the  liquid  boils.  It  is  at  once 
evident  that  the  boiling-point  will  be  lower  than  that  of  either 
pure  liquid. 

The  partial  pressures  in  the  vapour  are  in  the  ratio  of  the 
molecular  concentrations  : 


MI  +  na      HI  +  »2 

If  mi,  /»2  are  the  molecular  weights,  the  ratio  of  the  actual 
weights  of  the  components  distilling  over  is 

ici  :  wz  =  mini  :  /»2»2  =  PI»»I  :  Pa»»a       •          •     (1) 
an  equation  due  to  A.  Naumann. 

We  observe  that,  although  one  component  may  be  only  slightly 


410  THERMODYNAMICS 

volatile  alone,  it  will  distil  over  'freely  if  it  happens  to  have  a 
molecular  weight  which  is  large  in  comparison  with  that  of  the 
second  component. 

At  a  pressure  of  760  mm.  a  mixture  of  nitrobenzene  and 
water  boils  at  99°.  The  vapour  pressure  of  water  at  this 
temperature  being  733  mm.,  that  of  nitrobenzene  will  be 

760  —  733  =  27  mm.      Now  wi/>»2  =  18/123 
.'.  wi  :  «-a  =  18  X  733  :  123  X  27 
=  4:1,  approximately. 

Thus  one-fifth  of  the  distillate  is  nitrobenzene.  Since  the  com- 
plex organic  liquids  one  wishes  to  purify  are  usually  of  high 
molecular  weight,  the  method  of  distillation  in  steam  is  very 
valuable. 

If  nil  is  known,  and  PI  is  a  known  function  of  temperature, 
equation  (1)  serves  to  determine  m2,  the  molecular  weight  of  the 
second  liquid,  for  P2  is  determined  by  the  total  pressure  (usually 
1  atm.)  : 


177.     General  Theory  of  Binary  Systems. 

In  §§  163—  167  we  have  deduced  some  properties  of  systems  of 
two  components  in  two  phases  ("  binary  systems  ")  directly  from 
the  fundamental  principles,  and  in  §§  169  —  173  we  have  obtained 
quantitative  relations  in  certain  special  cases.  Here  we  shall 
obtain  some  general  equations  relating  to  such  systems  with  the 
help  of  the  thermodynamic  potential  (cf.  §  155). 

We  shall  first  establish  some  relations  which  apply  generally 
to  systems  of  n  components  in  r  phases.  Let 

iiti,   ))iz,  nia    .   .  .   ,  m,,' 
mi",  w»a",  •»!„",  .  .  .  ,  mn" 


be  the  masses  of  the  first,  second,  .  .  .  n-tli,  components  in  the 
first,  second,  .  .  .  r-th  phases. 

The  state  of  physical  and  chemical  equilibrium  of  the  system, 
at  a  constant  temperature  and  pressure  in  all  parts,  may  now  be 
completely  characterised  by  two  sets  of  relations  : 


GENEEAL  THEORY  OF  MIXTURES  AND  SOLUTIONS   411 

(i.)  In  each  separate  phase  the  chemical  potentials  of  the 
components  Ab  A2,  A3,  .  .  .  Are  must  satisfy  the  equation  : 

V\P\  +  V-2p2  +      •    •     •     +  vnPn  =  °    •  •  •       (1) 

where  j-iAi  -f  i-2A2  +    .  .      +  z'»Aw  =  0 

is  the  chemical  equation  representing  a  possible  reaction  between 
the  various  components  (cf.  §§  143,  155). 

(ii.)  The  chemical  potentials  of  each  component  must  be  the 
same  in  all  the  r  phases  of  the  system  : 

Pi  =  Pi"  =  pi"  =  •  •  •  =  pir\ 

p2  =  p2n  '  =  PS"  =  •  •  •  =  p2r\  _         •     (2) 

Pn     =  Pn"  =  Pn"  =     .    .    .    =  Pn' 

where  the  indices  denote  the  various  phases. 

If  the  system  undergoes  a  virtual  change  of  composition,  B,  at 
constant  temperature  and  pressure,  it  is  transferred  to  another 
state,  which  need  not  however  be  an  equilibrium  state,  since  the 
changes  of  the  masses  are  quite  arbitrary.  They  must,  however, 
satisfy  the  equations  : 

0  =  Snii   -j-  Bmi"  -\- 

0  =  8/»2'       8;wa"  .  .  . 

(6) 


since  the  total  mass  of  each  component,  no  matter  how  it  is 
distributed,  must  remain  constant.    We  have  also  : 

ST  =  0  ;    Bp=  0  ...     (4) 

Since  the  initial  system  was  by  hypothesis  an  equilibrium 
state  the  variation  of  potential  in  the  virtual  change  must  vanish 
to  the  first  order  : 

S</>  =  0    ......    (5) 

and  be  positive  to  the  second  order  if  the  equilibrium  is  stable  : 
82<£>0     .....     (6) 

Now  suppose  that  an  actual  change,  in  which  the  system  is 
transferred  from  one  equilibrium  state  to  another  infinitely  close 
equilibrium  state,  occurs,  and  let  us  denote  this  real  change  by  d 
to  distinguish  it  from  the  virtual  change  B.  The  temperature 


412  THEKMODYNAMICS 

and  pressure  will  also  change  along  with  the  composition  so  as 
to  keep  the  system  in  equilibrium.  Since  the  displaced  system 
is  also  in  equilibrium,  we  have,  for  a  virtual  displacement  from  it  : 

8(0  +  <ty)  =  0         .....    (7) 

/.  from  (5)  and  (7)  : 

&ty=0        .....     (8) 


dT  +       dp  +  2         dm,'  +          </,«,'  +  .  . 
3T  9p    7  3m/  dmz 

where  the  summation  extends  over  the  r  phases,  as  in  §  158. 
But  =-S,     ?*  =  V, 


p  m 

.•.  dQ  =  —  SrfT  +  YJp  -f-  "S,  fii'dmi   +  fjq'dmz  -\-   .  .  . 
.-.  8rf0  =  -  SSrfT  +  SVc/jj  +  ZSpidmi  +  S^'dm*   +  .  .  . 

But  aS  = 


where  W  is  the  heat  function  at  constant  pressure, 

m2'  +  .  .   =0  (9) 


an  equation  which  contains  the  laws  of  displacement  of  equili- 
brium of  a  system  in  the  most  general  form.  It  is  due  to  Willard 
Gibbs  (1876,  Scientific  Papers,  I.  ;  cf.  Duhem,  Traite,  I.,  ch.  viii.  ; 
III.,  ch.  i.,  ii.  ;  Planck,  Thermodt/namik,  3  Aufl.-,  §  211).  We 
observe  that  the  influence  of  temperature  and  of  pressure  vanish 
with  SW  and  SV  respectively.  8W  represents  the  heat  absorbed 
in  the  virtual  change,  and  SY  the  increase  of  total  volume. 
If  the  system  consists  of  one  component  in  two  phases  : 

wl  =  TSV  (claPeyron-clausiu^ 

since  S/V,   8^"  are  both  zero  on  account   of    the    unchanging 
composition. 

If  the  system  consists  of  the  three  phases  with  two  components  : 

solid  salt  +  solution  +  vapour 
we  obtain  Kirchhoffs  equation  (cf.  §  169). 


GENERAL  THEOEY  OF  MIXTURES  AND  SOLUTIONS    413 

The  case  which  interests  us  at  present  is  that  of  a  binary 
system,  i.e.,  two  components  in  two  phases  : 

=  ' 


+  S^i/m,"       .        .        .     (10)- 
Such  a  system  is  formed,  for  example,  by  a  binary  liquid  mix- 

ture in  contact  with  its  vapour,  or  a  mixed  crystal  in  contact 

with  its  melt. 

Let  -s'  =  mt'jmi,  s"  =  iwa"/»iilf  -         .     (11) 

be  the  concentrations  in  the  two  phases,  then  the  state  of  the 

system  can  be  expressed  in  terms  of  any  two  of   the  variables 

1>,  T,  s',  s". 

Now  5/V  =  §£l  S/in'  +  9^1  Sm/| 

.         .     (12) 


and  similarly  for  the  second  phase. 

For  the  given  system  there  are  two  kinds  of  virtual  change 
possible,  according  as  the  first  or  second  component  passes  from 
the  first  to  the  second  phase.  In  the  first  case  : 

Snii  =  —  Bmi"  ;    Sw.2'  =  —  &«a"  =  0 
and  in  the  second  case 

8ni  i  —  Bmi"  =  0  ;  8)11-2'  =  81)1-2". 
Hence,  in  the  first  case  : 

£  W 


and  in  the  second  case  : 

8W2"'  -  S*a"   =  0 


.        ._ 

77  =  l  1  »     5^  -  T/  —    *  2 


414  THERMODYNAMICS 

for  the  amounts  of  heat  absorbed,  and  the  increases  of  volume  per 
unit  transfer  of  the  first  or  second  component,  respectively,  from 
the  first  to  the  second  phase, 

.'.  4l  dl  -  rrfp  +  (V  -  S/ii")  =  0) 

...     (13) 
y  rfT  -  vydp  +  (S/*a'  -  S/V')  =  OJ 

We  now  calculate  the  values  of  the  expressions  in  brackets. 
From  (12) :  « 


To  change  the  independent  variables  ?»x',  ??ia',  ?n/',  ™2"  to  5',  «"  we 
have  the  relations  (11)  and  the  following,  derived  from  them  by 
partial  differentiation  : 


from  which  we  readily  find : 

**"*    !  =  ^-^&  (U) 


In  §  157  we  have  obtained  the  equations  : 

¥+8'¥=°=¥+8"¥= 

and  if  the  equilibrium  is  stable,  the  inequalities  : 


--<>or3ap-.-5>« 

s  9s"  s" 


.     (16) 
.        .    (17) 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    415 
Equations  (13)  now  go  over  into  : 

2i  dT  —  vjdp  +  <rVs'  —  a"ds"  =  0  .  (18a) 

^dT-<Ylp-a',ds'-\-(J''lds"  =  0      .         .    (186) 

.L  S  8 

which  contain  the  whole  theory  of  binary  systems. 

If  one  of  the  components  is  absent  from  one  of  the  phases,  we 
must  put  ds'  =  0,  or  ds"  =  0,  and  the  equations  will  then  apply 
to  the  evaporation  or  solidification  of  solvent  or  solute  alone. 

Let  us  take  the  general  case  in  which  both  phases  are  binary 
mixtures,  and  suppose,  first,  that  the  system  consists  of  a  mixture 
of  two  liquids,  say  alcohol  and  water,  in  contact  with  the  vapour. 

178.     Fractional  Distillation. 

If  a  binary  liquid  mixture  is  contained  in  a  cylinder  under  a 
piston,  and  the  latter  slowly  raised,  evaporation  proceeds,  and 
the  quantities  and  compositions  of  liquid  and  vapour  gradually 
change.  According  to  the  conditions,  the  evaporation  may  pro- 
ceed at  constant  temperature  (p  variable),  or  at  constant  pressure 
(T  variable).  The  latter  case  usually  occurs  in  practice  (cf. 
§§163—164). 

(i.)  Isopiestic  Distillation  : 


-a"ds"  = 


Ffom  these,  by  eliminating  ds",  and  ds',  respectively,  we  find  : 

(Sb) 


ds'  s' 

d^=  _To-" 
ds"  ~       ~s"~ 

The  differentials  are  total,  since  at  constant  pressure  T  is  a 
function  of  the  composition  of  either  the  liquid  or  the  vapour 
alone. 


416  THERMODYNAMICS 

(ii.)  Isothermal  Distillation  : 

dp  T<r'        s'  —  s"  ds'  s'        TI  -4-  s"v-2 

•  •  -r>  =  —  —r  -  -      —»-  >  or  T-  —  ~~  jfr'  -  — /—      -     •  (80) 
ds  s         n  +  *"ra          dp  To-         *'  —  «" 

^.=    -^!          *'  ~  *"   •    ->r  d*"  =         J-        ri  +  'V«         «M 

<fo"  «"    '  ri  +  *'ra'        <fy>  To-"        s'  — s" 

by  a  similar  process  of  elimination. 
At  constant  temperature,  p  is  a  function  of  s'  or  s"  alone. 
In  all  cases  we  have  the  inequalities  : 

a-'  <  0,    a"  <  0 
Qi>0,    Q2>0 

hence  from   the   above   equations  we  can  readily    deduce    the 
following  theorems : 

(1)  Constant  Temperature  : 

(a)  If  the  composition  of  the  liquid  is  altered  so  that  the  con- 
centration of  a  component  is  increased,  the  concentration  of  that 
component  is  also  increased  in  the  vapour. 

(b)  If  to  the  liquid  a  portion  of  that  substance  is  added  which 
the  vapour  contains  in  the  greater  proportion,  the  total  vapour 
pressure  is  increased,  and  inversely. 

(c)  The  pressure  has  a  maximum  or  minimum  value  when  the 
liquid   and  vapour  have  the  same   composition.     (Theorem  of 
Gibbs  and  Konowalow.) 

(2)  Constant  Pressure  : 

(a)  If  the  concentration  of  the  liquid  is  increased,  so  also  is 
that  of  the  vapour. 

(b)  If  to  the  liquid  a  portion  of  that  substance  is  added  which 
it  contains  in  less  proportion  than  the  vapour,  the  boiling-point 
is  lowered. 

(c)  The  boiling-point  is  a  maximum  or  minimum  when  the 
liquid  and  vapour  have  the  same  composition. 

(d)  An  increase  of  temperature  without  alteration  of  composi- 
tion of  the  liquid  raises  the  total  pressure. 

(e)  An  increase  of  temperature  without  alteration  of  composition 
of  the  vapour  raises  the  total  pressure. 

The  various  relations  have  already  been  described  (§§  163, 167). 


GENERAL  THEOEY  OF  MIXTURES  AND  SOLUTIONS   417 

179.     Isomorphous  Mixtures;  Solid  Solutions. 

The  relations  apply  also  to  the  case  of  a  liquid  mixture  of  two 
substances  which  is  solidifying  to  a  homogeneous  solid  which 
contains  the  two  substances  in  proportions  depending  on  the 
composition  of  the  melt — a  so-called  solid  solution  or  mixed 
crystal  (§  138). 

If  two  salts  which  do  not  react  chemically  to  produce  a  double 
salt  (e.g.,  K2S04  and  A12(S04)3),  or  another  salt-pair  (e.g.,  NaCl 
and  CuS04)  are  brought  in  contact  with  a  quantity  of  solvent 
insufficient  for  complete  dissolution,  the  composition  of  the  solution 
is  independent  of  the  proportions  of  the  two  solids  and  is  definite 
at  a  fixed  temperature,  as  we  see  from  the  phase  rule : 
F  =  w  +  2-r  =  S  +  2-S  =  2 

.'.  when  T  and  p  are  fixed,  so  is  the  concentration. 

This  was  verified  by  Riidorff  (1873)  with  a  large  number  of 
salt-pairs ;  e.g.,  NH4C1  +  NH4N03,  KC1  +  XaCl,  NH4C1  +  BaCL2, 
Na2S04  +  CuS04.  With  other  salt-pairs,  e.g.,  K2S04  +  (NH4)2S04, 
Ba(N03)2  +  Pb(N03)2,  CuS04  +  FeS04,  the  concentration  was  not 
uniquely  determined  by  the  temperature,  but  depended  on  the 
relative  amounts  of  the  two  salts.  It  was  observed  that  the 
members  of  all  such  salt-pairs  were  isornorphous,  and  it  was 
shown  by  Roozeboom  (1891)  that  the  apparent  contradiction 
vanishes  when  the  solid  is  regarded  as  a  homogeneous  crystal  con- 
taining both  constituents  in  varying  proportions — an  isomorplious 
mixture.  Then  r  =  2  and  .'.  F  =  3. 

These  two  processes — solidification  of  a  melt  and  crystallisation 
from  a  solvent,  are  the  most  important  cases  in  wbich  solid 
solutions  appear. 

The  theory  of  dilute  solid  solutions,  in  which  one  component 
is  present  in  small  amount  only,  has  already  been  considered 
(§  138). 

The  general  theory  was  worked  out  by  Roozeboom  (Zeitschr. 
physik.  Chem.,  1899)  from  the  standpoint  of  the  theory  of  thermo- 
dynamic  potential.  The  equations  (2a,  I),  (3«,  b)  of  the  preceding 
section  apply  equally  well  to  the  present  case,  and  details  need 
not  be  given  here.  The  liquid  solidifies  at  a  constant  tempera- 
ture when  it  has  the  same  composition  as  the  solid  deposited — 
the  so-called  eutectic  point. 

The  description  of  the  various  types  of  freezing-point  curves 


418  THEKMODYNAMICS 

which  occur  —  a  subject  of  great  importance  in  the  study  of 
metallic  alloys  —  will  be  found  in  text-books  on  the  Phase  Kule, 
e.g.,  Findlay,  Phase  Ride,  2nd  edit,  1906,  pp.  173  et  seq.  (cf. 
also  Desch,  Metallography,  1910). 

189.     Freezing-Points  of  Solutions. 

Let  the  system  consist  of  a  binary  liquid  solution  in  equili- 
brium with  the  solid  form  of  one  substance  in  the  pure  state. 
We  have  then,  if  the  doubly-accented  symbols  refer  to  the  solid  : 

*"  =  0,  ih"  =  0 

The  second  equation  (186)  of  §  178  then  vanishes,  and  the 
first  takes  the  form  : 

fi  dT  _  i-dp  +  crtfo  =  0      .        .'       .     (1) 

in  which  the  suffixes  and  accents  are  omitted. 

According  as  we  put  <7T,  dp,  or  d»  equal  to  zero,  we  have  the 
equations  representing  the  alteration  of  pressure  required  to  keep 
a  solution  of  altered  concentration  in  equilibrium  with  ice  at  the 
same  temperature,  or  the  alteration  of  freezing-point  with  con- 
centration, or  the  alteration  of  freezing-point  of  a  given  solution 
with  pressure,  respectively.  Similar  equations  apply  when  the 
solid  is  the  pure  solid  solute,  e.g.,  a  salt  along  with  its  saturated 
solution. 

The  most  important  case  is  the  alteration  of  freezing-point 
with  concentration  at  constant  pressure,  when  : 


'3T\    .         To- 
-- 


In  general  Q  <  0,  i.e.,  heat  is  evolved  when  ice  crystallises  out  ; 
then  T  falls  with  increasing  s.  In  the  case  of  strong  solutions 
of  sulphuric  acid,  Q  may  be  positive,  and  the  freezing-point  would 
rise  with  increasing  concentration. 

If  the  temperature  is  constant  : 


then 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS   419 

ThU8  ef  and  c  have  alwavs  °PPOsite  signs,  hence  if  a  solution 
is  in  equilibrium  with  ice  at  a  given  temperature  and  pressure, 
and  if  the  pressure  is  increased,  it  will  be  necessary,  in  order  to 
maintain  equilibrium,  to  |^^®  the  concentration  of  the  solu- 
tion according  as  the  crystallisation  of  ice  from  the  original 
solution  |  ^^^  the  total  volume.  There  do  not  appear  to  be 
any  experiments  on  this  subject. 

The  equations  just  obtained,  and  those  relating  to  vapour 
pressures,  are  quite  general  and  apply  to  solutions  of  any  con- 
centration. Unfortunately  we  are  not  yet  in  a  position  to 
calculate  the  magnitudes  o-',  cr"  in  the  general  case,  although 
we  have  seen  in  §  158  that  the  form  of  the  chemical  potential  a, 

and  hence   ^  =  -p   as   a   function  of   concentration,  is  known 

when  the  latter  is  very  small.  J.  J.  van  Laar  (Seeks  Vortraq? 
iibcr  das  thermodynam.  Potential)  has  also  worked  out  the  theory 
of  vapour-pressure  and  freezing-point  curves  on  the  assumption 
that  the  mixtures  conform  to  van  der  Waals'  theory  of  binary 
mixtures,  according  to  which  the  mixture  obeys  the  characteristic 
equation  : 


in  which          a  =  (1  —  x 


where  x,  1  —  x  are  the  numerical  molecular  concentrations 
and  the  magnitudes  ai,  CM,  02,  &i,  &2,  are  constants.  He  makes^ 
however,  certain  assumptions  with  respect  to  the  specific  heats' 
which  do  not  appear  to  be  justified,  and  we  shall  not  enter  into 
detail  here. 

In  the  development  of  the  theory  of  freezing-points  for  very 
dilute  solutions  (Chap.  XL)  it  was  assumed  that  both  the 
change  of  total  volume  and  the  heat  absorption  on  further  dilution 
are  zero.  With  solutions  of  moderate  concentration  (say  up  to 
5N),  neither  assumption  is  true,  but  we  know  that  the  change  of 
volume  is  always  very  small,  and_possibly  negligible,  whilst  the 
heat  absorption  is  not  usually  of  such  small  magnitude.  It 

E  E  2 


420  THERMODYNAMICS 

therefore  appears  that  a  theory  developed  on  the  assumption  that 
the  volume  changes  are  negligible,  but  which  takes  account  of 
the  absorption  of  heat,  will  very  probably  agree  fairly  well  with 
the  relations  actually  observed.  Such  a  theory  was  developed 
simultaneously  and  independently  by  Dieterici  and  T.  Ewan 
(1894),  and  may  be  called  the  theory  of  the  freezing  points  of 
solutions  of  moderate  concentration. 

If  the  solution  is  in  equilibrium  with  pure  frozen  solvent  at 
the  temperature  T  we  have  : 


...         .    (2) 

where  the  symbols  have  the  significance  of  §  131. 

If  TO  is  the  freezing-point  of  the  pure  solvent,  (1)  and  (2)  are 
integrable  on  the  assumption  that  Ag,  Ae  are  constant  only  if 
(T0  —  T)  is  small.  If  this  is  not  the  case,  i.e.,  the  solution  is  of 
moderate  concentration,  we  must  write  : 

A,  =  (A,)O  +  J  (c,  -  c>rr  =  (A.)0  +  cc,  -  c.)T    .    (3) 

Ap  =  (A.)0  +  J  (C,  -  C,)rfT  =  (Ae)0  +  (C,  -  C,)T      .     (4) 

where  Cs,  C/,  Gp  are  the  molecular  heats  of  solid,  liquid, 
and  gaseous  solvent  under  constant  pressure,  and  these  are 
assumed  independent  of  temperature  over  the  range  (To  —  T) 
(cf.  below,  where  it  is  seen  that  the  effect  of  temperature  below  T 
is  not  involved). 

Equations  (1),  (2)  are  again  integrable  : 


The  arbitrary  constants  Ii,  I2  are  eliminated  by  the  relation 
p'  =  p  =  po  when  T  =  T0 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    421 
.  '  .     from  (5)  —  (8)  we  find  : 


Put  (T0  —  T)  =  t   the    depression    of  freezing-point,    expand 

T 

™  into  a  power  series,  and  retain  the  first  three  terms  : 

"'r  =  '"(1  +  f)=f-2-i  +  8T3       .         -        .     (10) 
From  (9)  and  (10)  : 


£.  -  L  r(A»)o-(A*)o+(c<-c,)To  _  d-  c.  * 

p'      RL  ToT  2       T2 


•  (ID 


But     (Ag)0  -  (Ae)o  +  (C,  -  C,)  To  =  (Ar)0  +  (C,  -  C.)  To 

=  A,o.         .     (12) 
the  latent  heat  of  fusion  at  T0, 

.    ln  P  _  i  r  A,?  _  c,  -o.  j     c,-c.  *2n        Q3 
y  "  R  LTOT         2     T2  ^  ~~3      f3J 

The  equation  for  the  osmotic  pressure 

....      (14) 


deduced  in  §  131  is  true  when  there  is  no  change  of  total 
volume  on  mixing  the  solution  with  extra  solvent,  a  condition 
which  is  very  approximately  fulfilled  even  with  fairly  concentrated 
solutions,  hence  we  find  for  the  osmotic  pressure  at  the  freezing- 
point  T  : 

P_,rv    d-c,  *  ,  c,  -c,  fi  (  . 

[To"     ~2~    "T4      ~3~  T"aJ        '         * 

To  calculate  P  at  any  other  temperature  we  integrate  KirchhofF  s 
equation  for  the  heat  of  dilution  : 

&  .....  <16> 

As  a  first  assumption  we  take  Q  independent  of  temperature, 
i.e.,  we  suppose  that  the  same  amount  of  heat  is  absorbed  when  a 


422  THERMODYNAMICS 

mol  of  solvent  is  mixed  with  the  given  solution  at  all  temperatures 
in  the  range  T0  —  T.     Then  : 


If  T  =  To  : 


Add  (13)  to  (17)  and  subtract  (18)  : 


whence    the  osmotic  pressure   at  TO,   the  freezing-point  of  the 
pure  solvent,  is  : 


P  _ 


Ewan  took  account  of  the  variability  of  Q  with  temperature, 
which  introduces  the  specific  heat  of  the  solution  ;  this  is  usually 
quite  negligible  within  the  range  of  validity  of  (14).  \^-\-  Q  is  the 
heat  absorbed  when  a  mol  of  ice  melts  and  the  liquid  mingles 
with  the  solution. 

If  we  take  Dieterici's  numbers  for  the  vapour  pressures,  and 
Roloffs  for  the  freezing-points,  of  solutions  of  potassium  chloride, 
we  can  calculate  the  osmotic  pressure  (Po)  from  the  two 
equations  : 


(a)  P0  =  RT0    In    -,         (vapour  pressures). 

\      2)  '  ° 

(b)  p.  =  r  [*A+£_^=C.«^+C.rC.  M,-|  ((l.eezingpoints). 

In  the  table, 

m  =  grams  KC1  per  100  grams  solvent, 

t    =  TO  —  T  =  depression  of  freezing-point, 

Q  =•  heat  of  dilution, 

p'  =  vapour  pressure  of  solution  in  mm.  Hg  at  0°  C. 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    423 


Also,  p  =  4*620  mm.  =  vapour  pressure  of  water  at  0°  C. 

To  =  273,  d  —  Cs  =  18  X  0-475  =  8'55  cal.,  A/0  =  18  X  80'3 

=  1445-4  cal. 
R  =  1-985. 


•. 

P0  Calories. 

m. 

t 

Q. 

P*. 

Gale.  (a). 

Calc.  (6). 

0 

0 

0 

4-620 

_ 

_ 

3-72 

1-667 

1-63 

4-546 

8-805 

8-80 

7-45 

3-284 

5-96 

4-472 

17-645 

17-55 

14-90 

6-53 

19-5 

4-326 

35-595 

35-18 

22-35 

9-69 

34-3 

4-190 

52-905 

52-64 

181.     Graphical  Representation. 

The  treatment  of  systems  in  which  hydrates  (or  compounds), 
double  salts,  etc.,  are  deposited,  or  of  ternary  systems,  proceeds 
on  the  same  lines  as  the  investigation  of  the  simpler  cases 
considered  in  the  preceding  sections. 

The  equations,  especially  in  the  case  of  ternary  systems,  are 
necessarily  more  complicated,  but  nothing  fundamentally  new 
appears.  We  shall  therefore  omit  all  the  analytical  theory 
of  such  cases. 

There  are,  however,  two  graphical  methods  which  have  been 
largely  used,  arid  since  the  relations  involved  are  quite  simple,  a 
short  description  of  one  of  them  may  be  given  here.  The  funda- 
mental theorems  underlying  both  are  contained  in  Gibbs's 
memoir,  where  a  short  account  of  the  graphical  representation 
was  also  given.  The  latter  was  then  extended  on  the  one  side 
by  van  Rijn  van  Alkemade  (1893)  and  Roozeboom  (1899),  and  on 
the  other  side  by  van  der  Waals  (1891).  The  theorems  assert 
that  a  system  of  coexisting  phases  at  a  given  temperature  and 

pressure   s^rjves  ^0  a^ain  a  state  of  equilibrium  in  which  the 
I  volume 

rthermodynamic  potentials  </>    Qf  ^  te    hases  haye  fch 

\free  energies  ^ 
least  possible  sum. 


424 


THERMODYNAMICS 


The  consideration  of  </>  leads  to  the  potential  diagrams,  that  of 
i//-  to  the  free  energy  surface. 

182.     The  Potential  Diagrams. 

We  consider  for  simplicity  systems  of  two  components,  and  take 
as  unit  quantity  of  a  phase  that  containing  x  mols  of  the  first 
component  A,  and  (1  —  x)  mols  of 
the  second  component,  B,  i.e.,  1  mol 
in  all.  We  take  a  horizontal  axis, 
and  erect  on  it  two  perpendicular 
ordinates  at  unit  distance  apart.  If 
compositions  are  taken  as  abscissae, 
potentials  as  ordinates,  the  potential 
and  composition  of  every  possible 
phase  will  be  represented  by  a  point 
in  the  plane  between  the  two  parallel 
ordinates  -r  =  0  and  x  =  1. 

If  we  join  the  points  representing 
two  such  phases  by  a  straight  line, 
and  if  we  imagine  1  mol  of  a  hetero- 
geneous complex  formed  of  p  mols  of 

the  first  and  q  mols  of  the  second  phase,  the  complex  will  be  repre- 
sented by  a  point  on  the  straight  line  with  the  co-ordinates. 

=  p  xi  +  qx2 


where  x\,  x%  are  the  compositions,  and  $1,  $2  the  potentials  per 
mol,  of  the  first  and  second  phases. 

(x,  </>)  is  the  centre  of  gravity  of  masses  p,  q  placed  at  the  points 
(#i»  </>i)>  (#2,  </>2)>  respectively. 

Different  systems  composed  of  ,rA  +  (1  —  #)B  are  represented 
by  points  on  a  vertical ;  one  system  can  pass  spontaneously  into 
another  lying  below  it,  and  the  stability  is  indicated  by  the 
relative  height  of  the  point.  The  most  stable  system  (<£  an 
absolute  minimum)  is  represented  by  the  lowest  point. 

Now  consider  the  case  of  two  substances  forming  a  homo- 
geneous solution,  say  water  and  an  anhydrous  salt  (NaCl),  and 
let  us  draw  the  potential  curve  for  an  assigned  temperature  and 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    425 


pressure.  Let  a,  ft  be  the  values  of  <j>  per  mol  of  liquid  A  and  B, 
respectively,  at  the  given  temperature  and  pressure  (the  salt 
must  usually  be  regarded  as  existing  in  a  supercooled  state).  The 
shape  of  the  curve  in  the  vicinity  of  a  and  /3  is  determined  by 
Gibbs's  expression  for  the  chemical  potential  of  a  component  in 
a  very  dilute  solution  (§  158). 

=  yu,B  =  L  -f-  InM.  x 

where  L,  M  are  functions  of  T  and  p. 

For  x  =  0,  ^  =  —  oo  ,  so  that  the  curve  touches  the  <£  axis 

at  a. 

Similarly  at  /3. 

The  potential  is  a  continuous  function  of  the  composition, 
hence  the  curve  must  have  the  form  shown  in  Fig.  80.  The 
stability  of  a  solution  is  determined 
by  supposing  it  to  separate  into  two 
phases  represented  by  two  points  on 
the  curve  on  opposite  sides  of  the  point 
P  on  the  curve  representing  the  solu- 
tion. The  value  of  </>  for  the  hetero- 
geneous complex  is  the  ordinate  of  the 
point  of  intersection  P'  of  the  vertical 
through  P  with  the  join  of  the  two 
points,  and  the  homogeneous  solution 
is  stable  or  not  according  as  P  lies 
below  or  above  P',  i.e.,  according  as 
the  curve  is  convex  or  concave  to  the 
x  axis  in  the  vicinity  of  P.  If  a  point 
A  is  taken  on  the  <£  axis  to  represent  the  potential  of  a  mol  of  solid 
salt,  and  a  point  B  to  represent  the  potential  of  a  mol  of  ice,  the 
ordinates  of  the  points  of  intersection  of  straight  lines  from  A  and 
B  to  the  curve,  with  the  latter,  represent  the  potentials  of  the 
heterogeneous  systems  made  up  of  salt  and  solution,  or  ice  and 
solution,  in  the  proportions  represented  by  the  ab'scissse.  The 
tangents  from  A  and  B  to  the  curve  correspond  with  the  least 
values  of  </>,  and  the  points  of  contact  denote  solutions  which 
exist  in  equilibrium  with  solid  salt  or  with  ice,  respectively,  at 
the  given  temperature  and  pressure.  If  the  two  tangents  coalesce, 


FlG  80 


426 


THERMODYNAMICS 


we  have  ice,  salt,  and  solution  in  equilibrium  at  a  given  tempera- 
ture and  pressure — the  so-called  eutectic  point.  If  the  points  A 
and  B  lie  above  the  points  a  and  /3  respectively,  no  tangents  can 
be  drawn,  and  the  solution  cannot,  at  the  given  temperature  and 
pressure,  exist  in  equilibrium  with  salt  or  ice,  respectively.  Thus, 
the  point  B  lies  above  the  point  0  for  all  temperatures  above  273° 
for  aqueous  solutions,  since  ice  can  never  coexist  with  a  solution 
if  the  temperature  is  greater  than  the  freezing-point  of  the  pure 
solvent  at  the  assigned  pressure.  The  points  A  and  a,  and  B 
and  ft,  coincide  at  the  melting  points  of 
the  solid  salt  and  ice  respectively.  The 
form  of  the  curve,  and  the  altitude  of  A 
and  B  change  with  rise  of  temperature. 

Now  consider  two  coexisting  liquid 
phases,  such  as  those  formed  from  water 
with  ether,  phenol,  benzoic  acid,  or 
salicylic  acid.  For  the  homogeneous  solu- 
tions the  curve  is  convex,  and  the  separa- 
tion into  two  layers  therefore  implies 
that  at  some  intermediate  part  (cf.  §  115) 
the  curve  is  concave,  whilst  for  x  =  0  and 
x  —  I  the  course  of  the  curve  must  be  that 
indicated  above.  The  complete  curve  has 
therefore  the  form  shown  in  Fig.  81.  All 
heterogeneous  systems  lie  on  the  double 
tangent  PQ  ;  P  and  Q  represent  the  two 
liquid  phases  coexisting  at  the  given  tem- 
perature and  pressure.  All  points  on 
the  curve  between  P  and  Q,  represent 

labile  homogeneous  states.  From  each  of  the  points,  A  and  B, 
two  tangents  may  in  general  be  drawn  to  the  curve,  so  that  solid 
salt  or  ice  maybe  in  equilibrium  with  either  of  two  solutions 
of  quite  different  composition  at  a  given  temperature  and  pres- 
sure. With  change  of  temperature,  the  positions  of  A  and  a, 
and  of  B  and  ft,  change,  and  at  a  certain  temperature  the  tangent 
from  A  or  B  may  touch  both  P  and  Q  simultaneously.  There  is 
then  equilibrium  between  three  phases  :  two  solutions  and  solid 
A  or  B.  Thus,  at  98°  C.  solid  benzoic  acid,  a  very  concentrated 
solution,  and  a  solution  of  moderate  concentration  are  in  equili- 
brium. (This  is  the  so-called  "melting-point  under  water;" 


GENERAL  THEORY  OF  MIXTURES  AND  SOLUTIONS    427 


in  reality  the  "fused  acid"  is  the  very  concentrated  solution, 
represented  by  P.)  If  P  and  Q  coalesce  with  rise  of  temperature, 
both  liquid  phases  become  completely  miscible,  and  we  have 
reached  the  "  critical  solution  temperature  "  (§  174). 

Lastly,  consider  the  equilibria  of  solutions  with  a  solid  hydrate. 
The  latter  will  have  a  characteristic  potential  represented  by  a 
point  H  in  the  plane. 

All  points  on  the  two  tangents  HRi,  HR2,  to  the  curve  of 
solutions  represent  heterogeneous  systems  composed  of  solid 
hydrate  in  contact  with  solutions.  If  the  curve  between  R!  and 
R2  is  convex  the  heterogeneous  systems  are  stable,  and  inversely. 
At  a  given  temperature  and  pressure  the  hydrate  can  be  in 
equilibrium  with  two  liquid  phases  of  dif- 
ferent composition,  one  containing  relatively 
more,  the  other  relatively  less,  salt  than  the 
hydrate.  With  rise  of  temperature  the  form 
of  the  curve  and  the  altitude  of  H  change ; 
H  approaches  the  curve,  and  at  a  certain 
temperature  meets  it.  When  this  occurs, 
the  hydrate  is  in  equilibrium  with  one  solu- 
tion of  its  own  composition,  and  the  tem- 
perature is  the  melting-point  of  the  hydrate. 
At  higher  temperatures  no  equilibrium  exists. 
From  these  considerations,  the  general  form 
of  the  concentration-temperature  curve  is 
easily  deduced.  It  is  cut  in  two  points  by  a 
perpendicular  to  the  T-axis  and  is  bounded  on  the  side  of  higher 
temperature  by  a  tangent  line  also  perpendicular  to  the  tempera- 
ture axis.  The  abscissa  of  this  point  of  contact  denotes  a  maximum 
temperature  which  is  the  melting-point  of  the  hydrate,  and  its 
ordinate  represents  the  composition. 

The  curves  of  the  hydrates  of  ferric  chloride,  investigated  by 
Roozeboom,  afford  an  excellent  illustration  of  this  type. 

According  to  Le  Chatelier  (1889)  the  maximum  on  the  freezing-point 
curve  corresponding  with  the  deposition  of  a  chemical  compound  is  the 
intersection  of  the  two  curves  which  represent  the  lowering  of  freezing-point 
of  each  component  by  addition  of  the  other.  The  tangent  therefore  changes 
its  inclination  to  the  composition  axis  discontinuously  on  passing  the  maxi- 
mum, and  the  gradient  of  the  tangent  at  that  point  is  indefinite.  Roozeboom, 
however  (1889),  insisted  that  the  tangent  at  the  maximum  is  parallel  to  the 
composition  axis.  Kiister  and  Kremann  (1904)  pointed  out  that  the  part  of 


FIG.  82. 


428  THERMODYNAMICS 

the  curve  in  the  vicinity  of  the  separation  of  the  compound  is  a  flat  maximum 
when  the  compound  is  in  dissociation— equilibrium  with  its  components. 
Then  by  addition  of  either  component  the  dissociation  is  forced  back  and  the 
melting-point  is  lowered  to  an  extent  less  than  that  calculated  from  the  per- 
centage of  added  substance.  The  more  dissociated  is  the  compound  in  the 
fused  state,  the  flatter  and  more  extensive  will  be  the  maximum.  If  the 
compound  is  wholly  undissociated  in  the  fused  state,  the  two  curves  meet 
each  other  sharply.  The  first  case  is  observed  with  most  mixtures,  the  latter 
with  pyridine  and  methyl  alcohol  (Aten,  1905). 

From  the  shape  of  the  maximum,  some  information  can  be  obtained  as  to 
the  extent  of  dissociation  of  the  compound  in  the  liquid  state;  thus  the 
maximum  on  the  freezing-point  curve  of  CaCl2  and  H2O  is  very  broad  and 
flat,  an  indication  of  the  almost  complete  dissociation  of  the  compound 
CaCl2  .  6H20  in  the  fused  condition  (Eoozeboom  and  Aten,  1905  ; 
Kremann,  1906). 

The  same  methods  apply  also  to  ternary  systems,  the  only 
difference  being  the  use  of  a  third  axis  in  the  definition  of  the 
composition,  and  the  consequent  substitution  of  surfaces  for 
curves. 


183.     Liquefaction  and  Evaporation  of  Mixtures. 

If  a  gaseous  mixture  of  0'41  vols.  carbon  dioxide  with  0*59  vols% 
chloromethyl  is  compressed,  whilst  the  whole  mass  is  kept  agitated 
by  a  stirrer,  a  peculiar  series  of  changes  can  be  observed  (Kuenen, 
1892).  After  a  certain  point,  a  droplet  of  liquid  appears,  and 
this  increases  in  volume  with  the  pressure,  more  and  more  of  the 
mixture  becoming  liquefied.  Contrary  to  what  one  would  expect 
at  first  sight  to  find,  the  liquefaction  does  not  go  on  progressively 
till  all  the  vapour  has  disappeared,  but  the  quantity  of  liquid 
reaches  a  maximum,  then  decreases,  and  at  a  still  higher  pressure 
vanishes,  so  that  the  system  is  again  wholly  gaseous.  With 
further  increase  of  pressure,  liquid  again  appears,  and  finally  the 
whole  is  liquefied.  The  thermodynamic  theory  of  this  so-called 
retrograde  condensation  will  be  found  in  Duhem's  Traite,  t.  IV., 
pp.  109 — 156, and  in  Kuenen's  valuable  monograph:  Vcrdampfuny 
und  Verflussigung  von  Gemisclien. 


CHAPTER  XV 

CAPILLARITY   AND   ADSORPTION 

184.     Surface  Tension. 

The  effect  of  surface  energy  on  the  properties  of  heterogeneous 
systems  has  been  considered  in  §  101.  A  surface  possessing  an 
amount  of  surface  energy  2  per  unit  area  behaves  as  though 
subjected  to  a  tension  which  acts  tangentially  to  the  surface  at 
every  point,  and  tends  to  separate  two  portions  of  the  surface 
meeting  in  a  line  of  length  I  by  a  force  o7 ;  the  force  per  unit 
length  is  a-,  and  this  is  called  the  surface  tension.  It  is  easily 
shown,  by  considering  a  displacement  of  one  side  of  a  rectangular 
element  of  the  surface,  that : 


or  that  the  surface  energy  per  unit  area  is  numerically  equal  to 
the  surface  tension  per  unit  length. 

The  dimensions  of  surface  tension  are  therefore  : 


For  the  majority  of  liquids  a-  varies  from  20  —  100  —  —  . 

The  surface  tension  of  a  film  is  independent  of  the  area 
of  the  film,  so  that  the  latter  may  be  stretched  to  any  extent 
such  that  its  thickness  does  not  fall  below  a  certain  limit  without 
altering  the  value  of  a.  The  limit  (about  10  ~  8  centimetres)  is 
the  range  of  action  of  the  molecular  forces. 

In  the  older  theory  of  capillary  action  (1',  developed  by 
Laplace  T.  Young  (1805),  Gauss  (1830),  and  Poisson  (1831),  no 
attention  was  paid  to  the  possibility  of  thermal  changes  attending 
the  alteration  of  surface  at  constant  temperature.  That  such 
changes  must  exist  was  first  demonstrated  by  Lord  Kelvin  (2) 
(1859),  and  the  theory  of  capillarity  was  developed  more  parti- 
cularly from  the  thermodynamic  standpoint  in  the  masterly 
treatise  of  Willard  Gibbs  (3)  (1876). 


430  THERMODYNAMICS 

Let  us  consider  a  surface  film  (say  a  soap  film,  which  however 
is  composed  of  two  films)  of  area  o>  and  having  a  surface  tension 
a-  (T)  at  the  temperature  T. 

If  the  surface  is  increased  by  do>  the  work  done  is  : 

8Ai  =  -  adat        (rfw  >  0) 

since  the  work  is  positive  when  to  decreases. 

If  the  temperature  of  the  film  is  lowered  to  T  —  dT  at  constant 

(o,  the  surface  tension  becomes  (a  —  -^  dT),  and  if  the  film  is  now 
allowed  to  shrink  to  its  original  area  the  work  done  is  : 

SA2  =  +  (a-  -  ~  dT)  <2u   '  .        .        .    (dw  <  0). 

The  work  done  in  the  cycle  is 

(8A)  =  8A2  +  SAi  =  -  ^  dT  do>      .         .'        .         .     (1) 

Unless  -jm  =  0,  this  will  differ  from  zero,  and  by  the   equa- 
c  t  i 

tion  of  maximum  work  (§  58)  if  lud-a>  is  the  heat  absorbed  in 
stretching  the  film  : 


an  equation  analogous  to  that  of  Clapeyron  (§  63).  lu  is  the 
latent  heat  of  extension  of  the  film. 

In  all  cases  it  is  known  from  experiment  that  the  surface 
tension  diminishes  with  increase  of  temperature  of  the  surface. 
(Bede  has  shown  that  this  is  the  case  even  when  the  fluids  in 
bulk  preserve  their  original  temperature.) 

Corollary  1.  —  A  surface  film  absorbs  heat  when  extended. 

Example.  —  In  the  case  of  water  (soap-tiltns)  it  is  found  that  at  the  ordinary 
temperature  d<r/d£  =  —  (about). 

Hence  T  ^  =  l   x  290  =  41  =  -1  X  a-. 


Thus  the  latent  heat  of  extension  in  dynamical  units  is  equal  to  about 
half  the  work  spent  in  producing  the  extension  (Kelvin  1859). 


CAPILLARITY  AND  ADSORPTION  431 

If  the  temperature  and  area  of  a  film  are  increased  by  </T,  du>, 
the  element  of  heat  absorbed  is  : 

8Q  =  <yfT  +  /„<*«  .        .        .     (3) 

The  increase  of  entropy  is  : 

rfS=,.^  +  tT  W 

and  the  increase  of  energy  is  : 

<?U  =  <\.rfT  -f  (/„  -f  o-)  rf«        .         .     (5) 
But  dS,  <7U  are  perfect  differentials,  hence  by  applying  Eulers 
criterion  (H.  M.  §  57)  we  find  : 


andfrom(5):  =  £•-  •        •        •    (7) 

from  which  pair  of  equation's  we  can  readily  deduce  (2). 

=  -  ....    (8) 


so  that  the  rate  of  increase  of  entropy  of  the  film  with  extension 
at  constant  temperature  is  equal  to  minus  the  rate  of  increase 
of  surface  tension  with  temperature. 

Corollary  3.  —  At  the  critical  point  the  surface  of  separation 
vanishes,  hence  o-  for  the  surface  liquid  I  gas  must  also  vanish, 

/.  at  this  temperature  \-       =  0  (a  minimum)  and  hence 


=  °  (a  maximum)    •        •        •     (9) 

The  change  of  surface  tension  with  temperature  in  the  case  of 
states  far  removed  from  the  critical  point  is  found  experimentally 
to  be  linear  (Frankenheim  (4>  1836)  : 

:.  ^  =  const.,  or  ^  =  0        .         .     (i0) 
/.  o-  =  A  +  BT. 
Since  o-  =  0  when  T  =T«  A  =  —  BT. 


A    very  remarkable  theorem   respecthig  the  constant  B  was 
found  empirically  by  Eotvos  (5).(1886).      Let  a  mol  of  a  liquid 


482  THEKMODYNAMICS 

be  formed  into  a  sphere  in  contact  with  the  second  fluid  used  in 
the  determination  of  a-  (usually  air).  This  will  have  a  surface 
S,,,,  and  if  VMI  is  the  molecular  volume  : 


2OT<r  is  defined  as  the  molecular  surface  energy,  and  if  we  multiply 
(11)  by  this  we  have  : 

o-ys  =  B'2W(T  —  T,)  =  A-(T  —  T«)     .        .     (12) 

Eotvos's  law  can  now  be  expressed  in  the  form  : 

The  constant  k  is  independent  of  the  nature  of  the  liquid. 

Corollary  4.  —  The  temperature  coefficient  of  the  surface 
tension  of  a  liquid  is  inversely  proportional  to  the  molecular 
surface. 

Further  investigations  by  Ramsay  and  Shields  <«>  (1893) 
showed  that  equation  (12)  is  not  strictly  accurate  ;  they  replaced 
it  by 

„.  y|  =  -ft  (r  —  -t)  .         .  J."    .         .     (13) 

where  T  =  TK  —  T  and  e  is  usually  about  6.  Even  now  the 
equation  is  valid  only  if  T  is  greater  than  35,  i.e.,  T  is  fairly 
widely  removed  from  the  critical  point. 

The  mean  value  of  />;  for  a  large  number  of  liquids  is  2*12  ; 
liquids  containing  hydroxyl  groups,  or  those  which  have  an 
enolic  modification,  liquid  chlorine,  fused  salts,  and  metals,  have 
much  smaller  values  of  k,  and  this  is  attributed  to  polymerisation, 
since  if  the  actual  molecular  weight  is  larger  than  that  assumed 
in  calculating  Vl,  the  latter  value  will  also  be  smaller.  Ramsay 
and  Aston  (1894)  assumed  that  it  will  be  proportionally 
smaller,  but  the  values  of  the  polymerisation  coefficients  thereby 
deduced  obviously  depend  on  the  formula  assumed  for  the 
polymerised  molecule,  and  are  therefore  arbitrary. 

Again,  if  we  differentiate  (2)  with  respect  to  T  and  compare 
with  (6)  and  (10)  we  find,  in  the  case  of  states  far  removed  from 
the  critical  state  : 

^         _  L  dL?  +  '»         _  I  8('"  -  o 

<rr2  "      T  aT     T2         T  a»         . 

dc 

•••-5=  =  °    '     •     ~     •  (U) 


CAPILLARITY   AND  ADSORPTION  433 

This  implies  that  the  specific  heat  is  independent  of  the  area 
of  the  surface,  so  that  we  need  not  consider  any  special  heat 
capacity  belonging  to  the  surface  itself.  The  interpretation  is 
that  surface  energy  is  potential  energy,  not  kinetic  energ3r. 

If  we  compare  (14)  with  (7)  we  find : 

.         .         .     (15) 

But  (7M  +  <r)  is  the  rate  of  increase  of  the  intrinsic  energy 
with  the  surface  at  constant  temperature ;  equation  (15)  shows 
that  this  is  independent  of  temperature. 

In  the  vicinity  of  the  critical  point  (10)  is  no  longer  true,  and 
we  must  admit  the  possibility  of  a  special  heat  capacity  of  the 
surface  layer. 

Laplace  had  previously  deduced  from  his  theory  that  the  tem- 
perature coefficient  of  surface  tension  should  stand  in  a  constant 
ratio  to  the  coefficient  of  expansion ;  this  is  in  many  cases 
verified,  and  shows  that  the  effect  of  temperature  is  largely  to 
be  referred  to  the  change  of  density  (Cantor,  1892). 

Further  theoretical  investigations  will  be  found  in  van  der  Waals, 
Thcnnvdynamik,  I.,  218.  The  subject  has  also  been  developed  from  the 
kinetic  standpoint  by  Bakker,  Zeitschr.  pltysikal.  Chcm.,  28,  708,.  1899;  OS, 
684,  1910. 

Whittaker  V  (1908)  found  that  the  surface  energy  2  could  be  calculated 
from  the  internal  heat  of  evaporation  per  mol  (pM,  §  91)  by  the  equation  : 
2  =  K.T.  (PM) 

According  to  Kleeman  (1909)  the  constant  K  has  the  value  : 

K  =  0-557^ 
where  fIKt  T*  are  the  critical  density  and  temperature. 

185.     Adsorption. 

When  two  homogeneous  phases  are  placed  in  contact,  it  is  often 
found  that,  although  no  chemical  action  results,  a  more  or  less 
marked  alteration  of  concentration  occurs  at  the  interface.  Thus, 
gases  are  absorbed  by  porous  charcoal,  and  with  such  tenacit}'  is 
this  layer  of  gas  held  that  the  removal  of  the  minute  amount  of 
gas  remaining  in  a  space  exhausted  by  a  mercury  pump  is  best 
effected  by  means  of  cocoa-nut  charcoal  cooled  in  liquid  air 

T.  F  F 


434  THERMODYNAMICS 

(Sir  J.  Dewar).  The  removal  of  dissolved  colouring  matters  by 
charcoal  is  a  well-known  method  of  purifying  organic  substances. 
This  change  of  concentration  probably  occurs  whenever  a  surface 
of  separation  exists,  although  it  may  often  be  exceedingly 
small.  The  alteration  of  concentration  is  called  adsorption ;  if 
the  substance  increases  in  concentration  in  the  vicinity  of  the 
surface  the  adsorption  is  positive,  if  it  decreases  it  is  negative.  The 
equilibrium  is  found  to  be  attained  exceedingly  rapidly.  Thus 
if  a  gas  is  admitted  to  charcoal  which  has  previously  been  strongly 
heated,  the  pressure  falls  at  once  to  the  value  corresponding  with 
the  withdrawal  of  the  adsorbed  amount ;  it  afterwards  falls  again 
very  slowly,  an  effect  doubtless  due  to  the  slow  diffusion  of  the 
adsorbed  layer  on  the  surface  into  the  interior  of  the  mass 
(McBain  <8)  1907). 

The  adsorption  of  gases  on  solid  surfaces  proceeds  to  such 
an  extent  that  approximately  10 ~7  gr.  is  present  per  cm.2  in 
the  equilibrium  state.  This  is  of  the  same  order  of  magnitude 
as  the  strength  of  the  limiting  capillary  layer  of  a  liquid  (§  184), 
hence  it  is  not  improbable,  as  suggested  by  Faraday  (9>  (1834), 
that  the  adsorbed  gas  is  sometimes  present  in  the  liquid  state. 
The  adsorbed  amount  increases  with  the  pressure  and  diminishes 
with  rise  of  temperature.  The  first  effect  does  not  follow  a  law 
of  simple  proportionality,  as  in  the  case  of  the  absorption  of 
gases  by  liquids,  rather  the  adsorbed  amount  does  not  increase  so 
rapidly,  and  the  equation  : 

i  =  v*        .     .     .     .  a) 

where  x  =  total  mass  adsorbed  on  the  surface  m 

p  =  pressure 
a,  n  =  constants 
holds  good  (Freundlich  (10^  1906). 

If  n  =  1,  the  adsorbed  amount  would  be  proportional  to  the 
pressure;  in  adsorption  phenomena  n  >  1. 

In  the  case  of  solutions,  if  we  define  the  extent  of  adsorption 
in  the  same  way,  the  equation 

£  =  «<*       •'      •         •         •         r..'« 
applies,  where  £  is  the  concentration  of  the  solution  (Freundlich 


CAPILLARITY  AND  ADSORPTION 


435 


1906).     u  is  always  greater  than  unity ;  it  may  reach  the  value 
10—12. 

It  is  of  course  assumed  that  no  chemical  reaction  occurs 
between  the  adsorbent  and  the  substance  adsorbed;  if  this 
is  the  case  (as  may  happen,  for  example,  in  the  taking  up  of  a 
dye  by  a  fibre),  the  equation  (2)  no  longer  holds  good,  and  the 
solute  may  be  practically  completely  withdrawn  at  all  concentra- 
tions. 

If  we  suppose  that  a  solution  A  of  concentration  £  is  in  con- 
tact with  an  immiscible  substance  B  (say  air,  or  petroleum)  over 
a  surface  S,  there  will  be  a  different  concentration  of  the  solute 
in  the  immediate  vicinity  of  S  from  that  in  the  free  bulk  of  A. 

This  generalisation  is  due  to  Gibbs  (1874),  who  at  the  same 
time  showed  how  quantitative  relations  could  be  found. 

We  express  the  altered  concentration  in  terms  of  the  adsorption 
excess.  If  all  the  adsorbed  substance  were  contained  to  the 
extent  of  k  gr.  per  cm.2  on  a  superficial 
layer  of  zero  thickness  and  surface  co, 
the  total  mass  present  in  the  volume  Y 
would  be  m  =  Y£  +  /,•<•>.  The  layer  of 
altered  concentration  must,  however, 
have  a  certain  thickness.  We  will  there- 
fore imagine  a  plate  2  placed  in  front 
of  the  surface  and  parallel  to  it,  and 
define  the  adsorption  excess  as  the  concentration  in  the  included 
layer  minus  the  concentration  in  the  free  liquid.  That  this 
result  is  independent  of  the  arbitrarily  chosen  thickness  is  easily 
proved  when  we  remember  that  the  problem  is  exactlv  the 
same  as  that  of  finding  the  change  of  concentration  around 
an  electrode  in  the  determination  of  the  transport  number  of  an 
ion  by  Hittorf's  method. 

Let  us  suppose  a  quantity  w  gr.  of  solute  to  wander  into  a 
layer  of  thickness  S  parallel  to  the  adsorbing  surface.  If 
the  area  of  the  latter  is  eo,  the  adsorption  excess  is  defined  as 

f  —  — ,  i~C;  the  excess  per  cm.2  of  surface. 

The  layer  B  we  shall  call  the  true  adsorption  thickness  ;  the 
liquid  beyond  S,  which  has  almost  exactly  the  original  concen- 
tration provided  a  large  volume  was  present  at  the  start  and 
T  is  small,  we  shall  call  the  free  liquid.  Now  suppose  a 

F  F  2 


436 


THERMODYNAMICS 


layer  of  thickness  d  is  isolated,  and  the  liquid  contained  in  it 
mixed  and  analysed.  The  layer  d  we  shall  call  the  arbitrary 
adsorption  thickness.  If  £  is  the  concentration  of  the  free 
liquid,  the  amount  of  solute  in  the  arbitrary  layer  is  £««/  -f-  «-, 
whereas  that  in  a  similar  layer  of  free  liquid  is  £«*/.  The 

excess  is  ir,  and  the  excess  per  cm.2  is  -,  i.e.,  T. 


a  a' 


186.     Gibbs's  Adsorption  Formula. 

In  his  original  demonstration  Gibbs  (1874)  showed  that 
the  surface  layer  may  be  considered  as  a  third  phase  having 
specific  values  of  density,  energy,  and  entropy,  and  further 
that  the  results  of  the  theory  are  quite  independent  of  the  actual 
extent  of  the  capillary  layer  and  the  way  in  which  it  merges  into 
the  free  fluids  on  either  side.  As  a  matter  of  fact,  the  transition 

of  density,  etc.,  probably  occurs 
continuously  but  rapidly  over 
a  very  small  distance  (van  der 
Waals). 

Let  us  suppose  that  we  have 
a  solution  A  in  contact  at  one 
side  with  a  surface  of  adsorp- 
tion  ab  separating  it  from 
another  phase  B  which,  for 

simplicity,  we  shall  first  take  to  be  the  vapour  of  the  solvent.  At 
the  other  side  the  solution  is  in  contact  with  pure  liquid  solvent 
C  through  a  semipermeable  piston  c,  exposed  to  an  osmotic  pres- 
sure P. 

Let  the  volume  of  the  solution  be  V,  its  concentration  £,  and  let 
the  area  of  the  adsorbing  surface  ab  be  <a. 

(i.)  Let  ab  be  increased  from  «  to  o>  -}-  do)  whilst  V  remains 
constant  (say  by  alteration  of  form  to  a'b').  The  work 
done  is 

&Ai  =  —  a  da> 

where  o-  is  the  surface  tension  at  the  interface. 
At  the  same  time  the  osmotic  pressure  changes  from  P 


FlG 


(ii.)  Let   the  volume  of  the  solution  be  increased  from  V  to 


CAPILLARITY  AND  ADSORPTION  437 

y  _j_  <JY  whilst  the  surface  a'b'  is  unchanged.    The  work 
done  by  the  piston  c  is 


whilst  the  tension  alters  from  a-  to  (  <*  +  ^^  )  • 

(iii.)  Let  the  surface  be  restored  to  its  original  area«.     The 
work  done  is 


«».=  (*  4- 

whilst  the  osmotic  pressure  regains  the  value  P. 
(iv.)  Let  the  volume  be  decreased  by  dY,  when  the  initial  state 
is  restored.     The  work  done  is  : 


An  isothermal  reversible  cycle  has  been  executed 

.-.  2A  =  0 
/.  SAX  +  SA2  +  SAu  +  8Ai  =  0 


.  a, 


The  concentration  (and  therefore  the  osmotic  pressure)  of  the 
solution  depends  on  the  extent  of  the  surface.     The  definition  : 


applies  ta  solutions  which  are  homogeneous  in  bulk  ;  in  the 
present  case,  however,  there  is  an  ad$orpti*m  rxcm*  of  T  mols 
per  unit  area  of  the  surface  : 

::(  =  '-=  ....    (2) 


Thence,  since  (1)  can  be  written  : 

^«r      Zg  rP     eg 

ae-w^-jf-s 

and  we  have,  by  differentiation  of  (2)  : 


438  THERMODYNAMICS 

and : 


da-  3P 

=  W  Z  8?          •'•      '        '     (4) 


From  (la),  (2a),  and  (8)  we  find  : 

*  ^(T  (/(T 

d£  —  8/w£ 
If  the  solution  is  dilute  :   P  =  £RT (5) 

•••fi^lly  r=-OT'|     '  <6> 

According  as  —  is  =  0  there  will  be  positive,  zero,  or  negative 

adsorption,  respectively. 

The  method  of  deduction  also  applies  when  B  is  a  liquid,  or  a 
solid,  and  (6)  therefore  holds  for  these  cases.  The  equation  (6)  is 
called  Gibbs's  Adsorption  Formula  ;  it  was  deduced  independently 
by  J.  J.  Thomson  <»>  (1888).  The  present  deduction  is  due  to 
Milner  W  (1907). 

187.     Experimental     Examination     of     Gibbs's      Adsorption 
Formula. 

On  account  of  the  very  great  difficulty  of  measuring  the 
extremely  small  amounts  of  adsorbed  substance  at  a  liquid/yets 
or  liquid/liquid  interface,  very  few  experiments  are  available  for 
testing  Gibbs's  equation.  Zawidski (13)  (1900)  pointed  out  that 
the  concentration  of  the  foam  of  a  solution  should  be  different 
from  that  of  the  latter  in  bulk,  and  Miss  Benson (14)  (1903)  by 
the  analysis  of  a  solution  of  amyl  alcohol  in  water  found 

in  the  foam   0'0394  mol  per  litre 
„    „    liquid  0-0375     „      „     „ 
.*.  difference   =  0'0019     „      „     „ 

A  quantitative  examination  of  this  case  has  been  made  by  R. 
Milner  (1907),  W.  C.  McC.  Lewis,  ^  and  Donnan  and  Barker  <16) 
(1911). 

The    latter   passed    air   bubbles    up    a    tube    containing   an 


CAPILLARITY  AND  ADSORPTION 


439 


aqueous  solution  of  nonylic  acid,  or  of  saponin.  Each  bubble 
carried  with  it  a  very  slight  alteration  of  concentration,  and  after 
a  time  the  concentrations  at  the  top  and  bottom  of  the  tube  were 
different. 

F  (nonylic  acid)  =  1  x  10~7          (agrees  with  calcd.). 


T  (saponin) 


=  4  X 


1(T7  -        (twice  as  great  as  calcd.). 
cm* 

In  the  case  of  liquid  /liquid  interfaces  we  have  the  experiments 
of  W.  C.  McC.  Lewis  (1908),  who  examined  the  relations  at  the 
surface  of  separation  between  an  aqueous  solution  and  paraffin 
oil  or  mercury.  If  o-,  a-'  are  the  surface  tensions  between  paraffin 
oil  and  pure  water  and  the  solution,  respectively,  it  was  found 
that  o-'<  o-,  i.e.,  the  substances  examined,  were  positively  adsorbed. 

The  adsorption  excesses  of  the  ions  were  calculated  according 
to  §  203.  Substances  were  divided  into  three  classes  according  as 

the  adsorption  excess  F  -^-^  was  about  60  times  (Na  glycocholate 

cm.  , 

r  =  7  X  10"s  calc.  ;  5  X  10~;;  obs.),  or  5—10  times  (Ag 
4-5  X  10"9  calc.  ;  2'5  X  10~8  obs.  ;  in  AgN03),  or  about  equal 
(Caffeine  2'4  X  10~8  calc.  ;  3'7  X  10~s  obs.)  to  the  calculated. 

The   deviations   are   all   on  the  same  side,   viz.,    the    actual 
adsorbed   amount   is  too  great.     Further 
investigation  will  probably  show  what  is 
the  cause  of  this  phenomenon. 

If  we  now  enquire  what  sort  of  influence 
different  solutes  have  on  the  surface  ten- 
sion of  a  liquid  in  contact  with  its  vapour, 

i.e.,  the  magnitude  of  ~r,  we  find  that,  in 
dc 

the  case  of  water,  solutes  may  be  divided 

into  two  classes,  the  members  of  which 

exert  either  a  very  strong  influence,  or  a 

very  slight  influence,  respectively,  on  the 

surface  tension.    The  former,  called  active 

substances,  include  the  halogens,  fatty  acids 

(especially  higher   members),  alcohols,   aldehydes,  amines,  and 

esters.  ;17)     Thus,  in  a  0*00079  normal  solution  of  nonylic  acid  the 

surface  tension  of  water  is  lowered  from  75*3  to  40.     The  other 

class  of  inactive  substances  includes  the  salts  of  inorganic  acids. 


FIG. 


440  THERMODYNAMICS 

It  is  curious  that  active  substances  are  usually  strongly  smelling. 
In  the  typical  diagram  wre  see  that  the  addition  of  B  strongly 
lowers  the  surface  tension  of  A  (Fig.  85). 

The  adsorption  equation  shows  that  a  solute  may  very  strongly 
lower  the  surface  tension  of  a  solvent,  but  cannot  strongly  raise 
it,  since  although  F  may  reach  high  values  by  positive  adsorption 
(in  some  cases,  as  with  solutions  of  some  aniline  dyes,  the  pure 
solute  appears  as  a  thin  skin  on  the  surface),  it  can  never  sink 
below  that  of  the  pure  solvent  by  negative  adsorption. 

In  the  case  of  inactive  substances  the  difference  between  the 
tensions  of  solution  and  solvent  is  very  nearly  proportional  to 
the  concentration  (18^  : 

With  active  substances  it  is  proportional  to  some  power  of  the 
concentration  ^  : 

S  W  **9  »».»•  \^J 

where  *,  -  are  constants.     The  value  of  -  is  very  similar  for 
n  n 

different  substances.  If  we  combine  (2)  with  Gibbs's  equation 
we  obtain  Freundlich's  adsorption  equation  (§  185). 

The  tension  of  a  solution  is  lowered  with  rise  of  temperature 
according  to  a  linear  law  : 

<re  =  o-0(l  -  70)     .         .         .         .     (3) 

188.     Adsorption  of  Gases  on  Solids. 

In  considering  the  adsorption  of  gases  on  solid  surfaces  we 
will  suppose  that  m  gr.  adsorbent  take  up  x  c.c.  gas,  reduced  to 
N.T.P.  Then,  since  the  solid  adsorbent  has  a  spongy  structure, 

—  is  proportional  to  ar,  the  number  of  c.c.  adsorbed  per  cm.2 
surface.  Strictly  speaking 

T  =  x  -  f 

where /is  the  gas  present  per  cm.2  of  surface  when  no  adsorption 
occurs  ;  this  is  always  so  small  compared  with  x  that  we  can  put 

r  =  a; 

,.  ra£.     .       .       .       .    (4) 


CAPILLARITY  AND   ADSORPTION 
We  have,  from  the  adsorption  equation  : 


441 


.     (5) 


The  following  numbers  '•-'•'      refer   to   ammonia   adsorbed   on 
meerschaum  at  0°  C. 

a  =  54-83,  -  =  0-184 


p  cm.  Hg. 

.r  m  obs. 

x/m  calc. 

0-500 

48-3 

48-3 

3-713 

72-3 

69-8 

•21-500 

95-3 

96-4 

57-56 

117-0 

116-0 

-  depends  only  slightly  on  the  nature  of  the  gas  and  adsorbent, 

varying  from  0*2  to  0'6. 

The  constant  characteristic  of  the  adsorption  is  a,  the  number 
of  c.c.  gas  taken  up  under  1  cm.  pressure,  a  is  all  the  greater 
the  more  easily  liquefiable  is  the  gas. 

Further,  the  nature  of  the  gas  far  outpaces  that  of  the  adsor- 
bent in  its  influence  on  the  value  of  a,  a  fact  noticed  by  de 
Saussure  <21>  as  early  as  1814.  According  to  Baerwald  <->  (1907) 
if  A,  B,  C  .  .  .  denote  different  gases,  it  appears  that  the 
relation 

'•L'WCU. 


(-)=(-) 

\a   1  4     \a   1 


(3) 


holds   as   a  limiting  case,  where  a',  a"  are  the  constants    for 
charcoal  and  glass  respectively. 

The  same  relation  has  been  observed  in  adsorption  from  liquid 
solutions,  but  is  not  absolute. 

It  is  to  be  observed  that  the  value  of  ~  is  the  determining 

factor  in  adsorption ;  if  this  is  small  a  large  increase  of  surface 
will  produce  very  little  increase  of  adsorbed  amount— a  pheno- 


442 


THERMODYNAMICS 


menon  which  has  given  rise  to  the  erroneous  view  that  adsorption 
must  always  be  a  chemical  combination. 

In  the  adsorption  of  a  gas  on  a  liquid,  in  which  it  may  be 
more  or  less  soluble,  we  may  replace  the  concentration  by  the 
partial  pressure,  since  the  amount  dissolved  is  proportional  to 
the  latter: 


_ 
Rf  dp 


(4) 


In  the  adsorption  of  gases  on  solids  the  constants  a  and  - 
decrease  and  increase,  respectively,  with  rise  of  temperature. 

C02  on  Charcoal  (Travers  <-3)  1906) 


6 

a 

, 

n 

-  78 

14-29 

0-133 

0 

2-96 

0-333 

+  35 

1-236 

0-461 

61 

0-721 

0-479 

100 

0-324 

0-518 

In  considering  the  effect  of  temperature  we  have  two  important 
cases : 

(1)  p  is  kept  constant  and  the  adsorbed  amounts  at  different 


temperatures   are  compared.       The    curves      — ,  T )    are    called 

\m      ' 


Isopncuma  (Ostwald).  (21 
x 

in 


(2)       -    is    kept    constant    and    the  pressures    at    different 
temperatures   are   compared.      The   curves   (p,  T)?    are   called 

Isosteres. 

We  first  consider  the  isopneuma.     It  has  been  found  experi- 
mentally that,  for  a  given  pressure  p{ : 


(5) 


CAPILLARITY  AND   ADSORPTION  443 

where  5  is  a  decreasing  function  of  pressure  : 

/  JW  (*)  =  *»(*)-  (5  -#ni>)0.        .        .     (6) 
\m/  e        vTw/ 


or  *    =    -e-  *-*•*»          .         .         .  (6a) 

\w/  0     \m/  o 

In  the  case  of  isosteres  there  is  no  such  simple  relation. 

We  now  differentiate  the  adsorption  equation  (2)  with  respect 


The  equation  of  an  isopneumon  is  -^  =  0 


Now  differentiate  (6)  with  respect  to  0  at  constant  p  : 

•     -    •  » 


But  In  at  =  Ina0  —  5#  from  (6)  : 
.'.  from  (11)  and  (13) 


75 
Thence 


da  /-,  n\ 

^  —  s.     •     •     •  do) 


(ID 


_ 
'  ~W     ~  ° 

The  change  of  In  •'—  is  composed  additively  of  the  two  changes 
of  Ina  and  -  with  temperature. 


•]    /  X  \ 

The  equation  of  an  isostere  is  (  .1"    =  0. 


444  THERMODYNAMICS 

Divide  through  by  p,  and   substitute  the  values   of  (-^  and 

*<*), 

d0 

/7  7/i  111 

.         .        .         .     (12) 


Thus  — -*-   [a  no£  constant ;  it  increases  with  falling  tempera- 
du 

ture  or  pressure. 

— j0-  is  independent  of  the  nature  of  the  solid  adsorbent  over 
wide  intervals. 
189.     Heat  of  Adsorption. 

It  was  observed  by  de  Saussure  in  1814  that  heat  is  evolved 
during  the  adsorption  of  gases  on  charcoal,  and  quantitative 
measurements  were  made  by  Favre  (24)  (1874),  Chappuis  (1883), 
and  Dewar  <25>  (1904). 

According  to  the  circumstances  attending  the  adsorption  we 
may  have  different  heats  of  adsorption  : 

(1)  Integral  Heat  of  Adsorption — corresponding  with  a  heat 
of  solution,  and  evolved  when  the  gas  is  brought  in  contact  with 
just  enough  adsorbent  to  take  it  up. 

(2)  Differential  Heats  of  Adsorption — corresponding  with  heats 
of  evaporation  (§  173),  and  evolved  or  absorbed  when  one  equili- 
brium   state  |jPi>(~)    I  is   transferred   to   another  Ipa,  (— ) 

The  change  may  be  effected  : 

(a)  laoaterically — one  phase  has  an  unaltered  composition  and 

the  pressure  in  the  other  varies  r—  const. ;  p.  T  variable ) .     The 

\m  I 

heat  of  adsorption  is  analogous  to  the  heat  of  evaporation  of  a 
mixture. 

(b)  Isopneumically — the  composition     of    one    phase     alters 
whilst  the  pressure  in  the  other  remains  practically   constant 

p,  T  const. ;  —  variable).     The  heat  of  adsorption  corresponds 

with  the  heat  of  reaction  in  a  condensed  system. 

The  isosteric  heat  of  adsorption  is  determined  by  the  Clapeyron- 


CAPILLARITY  AND  ADSORPTION  445 

Clausius  equation.     If  the  initial  conditions  are  T,  -  ,  p,  and  the 
final  conditions  T  +  dT,     ,  p  +  dp,  it  is  readily  proved  that : 

Heat  of  adsorption  per  mol  =  q  =  RT2  (—j^ 
But  ^  =  ,,(5-. 


No  researches  with  the  proper  conditions  are  available.     The 
heat  of  adsorption  is  often  greater  than  the  latent  heat  of  evapora- 
tion (or  even  than  the  latent  heat  of  sublimation)  of  the  gas  in 
the  liquid  (or  solid)  state.     Thus  for  1  mol  XH3  on  charcoal : 
heat  of  adsorption  =  8100  cal. 
heat  of  sublimation  =  5000  cal. 

It  may  be  that  the  liquid  layer  is  strongly  compressed,  when  it 
would  have  a  higher  vapour  pressure  and  heat  of  evaporation. 

Similar  gases  have  similar  heats  of  adsorption  (C02,  XH3 
S02,  CHC13);  difficultly  liquefiable  gases  have  much  smaller 
values. 

190.     Wetting. 

If  a  drop  of  liquid  is  placed  on  a  solid,  the  condition  that  it 
spreads  over  and  wets  the  latter  is  : 

(T(s_g]><r(l_g}  +  <r(,_/( 

where  %-<,)  is  the  tension  between  eolid  and  gas,  and  so  on. 
This  is  readily  proved  by  considering  a  small  displacement  and 
putting  the  work  done  positive. 

Chappuis  (1883)  observed  that  many  inert  solids  evolve  heat 
when  moistened  with  water.  Thus  : 

1  gr.  wood-charcoal  evolves  7*4  cal. 
1  gr.  dry  clay  evolves  2'8  cal. 

when  just  soaked  with  water. 

Parks  (2t5)  (1902)  has  shown  that  the  heat  evolved  is  propor- 
tional to  the  surface.  Jungh  (27>  (1865)  and  Schwalbe  <2S>  (1905) 
made  the  remarkable  observation  that  the  heat  evolved  on 
wetting  a  solid  with  water  is  positive  below  4°C.  and  negative 


446  THERMODYNAMICS 

above  that  temperature.  It  is  therefore  probably  connected  with 
the  heat  of  compression  /,„  and  arises  from  the  strong  attraction 
of  the  solid  for  the  liquid. 

191.  Characteristics  of  Adsorption  Phenomena. 

According  to  Freundlich  (-9)  adsorption  equilibria  may  be  dis- 
tinguished from  chemical  equilibria  by  the  following  peculiari- 
ties : 

(1)  The  conformity  to  the  Adsorption  Isotherm  :  —  =  ag\ 

(2)  The  slight  variability  of  —  with  very  different  substances. 

(3)  The  slight  variability  of  a  with  the  nature  of  the  solid 
phase. 

(4)  The  very  large  velocity  of  attainment  of  equilibrium. 

(5)  The  relatively  small  displacement  with  temperature. 

192.  The  Phase  Rule  and  Dispersed  Systems. 

Modern  experiment  has  proved  beyond  doubt  that  the  so- 
called  colloidal  solutions  are  systems  composed  of  two  or  more 
phases,  i.e.,  heterogeneous,  characterised  by  an  enormously 
great  extent  of  division,  in  which  the  surface  of  contact  has, 
so  to  speak,  been  spread  out  throughout  the  whole  mass.  Capil- 
lary phenomena  are  therefore  predominant  here  (cf.  Ostwald, 
Kolloidchemie,  Leipzig,  1909  ;  Freundlich,  Kapillarchemie,  Leipzig, 
1909). 

A  surface  of  separation  between  two  phases  is  called  a  specific 
surf  ace  of  separation,  and  in  considering  the  states  of  such  systems 
it  is  evident  that  every  specific  surface  constitutes  a  new  inde- 
pendent variable.  If  there  are  n  components  in  ;•  phases  with 
x  specific  surfaces,  the  Phase  Rule  will  therefore  read : 
F  =  2  +  n  +  x  -  r 

A  classification  of  dispersed  systems  on  this  basis  has  been 
worked  out  by  Pawlow  (30)  (1910),  who  introduces  a  new  variable 
called  "  the  concentration  of  the  dispersed  phase,"  i.e.,  the 
ratio  of  the  masses  of  the  two  constituents  of  an  emulsion,  etc. 
When  the  dispersed  phase  is  finely  divided  the  thermodynamic 
potential  is  a  homogeneous  function  of  zero  degree  in  respect 
of  this  concentration. 


CAPILLARITY  AND  ADSORPTION  447 

193.     Surface  Tension  and  Solubility. 

The  influence  of  division  on  solubility  has  been  mentioned  in 
§  137.  According  to  Hulett  if  A,  Aw  are  the  solubilities  of  the 
substance  in  lumps  and  in  grains  of  radius  r  respectively, 


A        pr 

where  p  =   density,  a-  =  tension    at  solid/liquid   interface  (ef. 
Freundlich,  Kapittarchemie). 

194.  Influence  of  Capillarity  on  Chemical  Equilibrium. 

It  can  be  shown,  (Gibbs,  Scientific  Papers,  I.  ;  J.  J. 
Thomson,  Applications  of  Dynamics  to  Pliysics  and  Chemistry), 
that  a  chemical  equilibrium  can  be  modified  by  the  action 
of  capillary  forces.  Thus,  a  state  of  equilibrium  in  solution 
rua}*  conceivably  be  modified  if  the  latter  is  in  the  form  of  thin 
films,  such  as  soap  bubbles.  Since,  according  to  Freundlich 
(Kapillarchemie,  116),  there  is  at  present  no  direct  evidence  of 
the  existence  of  such  modification  (which  would  no  doubt  be 
exceedingly,  though  possibly  measurably,  small)  we  shall  not  enter 
any  further  into  the  matter  here. 

195.  Heat  of  Swelling  of  a  Colloid. 

If  gelatine,  or  starch,  etc.,  is  placed  in  water,  it  absorbs  the 
latter  and  produces  a  flabby  mass.  At  the  same  time  heat  is 
evolved.  This  is  supposed  to  result  from  the  work  done  by  an 
unknown  tension  P,  which  tends  to  open  out  the  structure  of  the 
colloid. 

At  T°  let  the  contraction  of  total  volume  =  A</>  ;  the  work  done 
is 


At  T  +  dT  the  tension  has  the  value  f  —  ^dT  +  ^-^ 

01  0<p  01 


l     J  •          T>  .  jrr       A 

...  work  done  is    P  -  -^  +  ^       «IT  A*. 

The  difference  of  .the  two  amounts  of  work  is  equal  to  the  heat 

im 

absorbed  multiplied  by  -^  : 

ap     ap    a<) 


448  THERMODYNAMICS 

The  expression  in  brackets  may  be  regarded  as  practically 
constant,  hence  the  heat  evolved  should  be  proportional  to  the 
contraction.  By  examining  the  swelling  of  starch  containing 
different  amounts  of  water,  in  water,  this  result  was  verified  by 
H.  Rodewald  (1897). (31) 

196.     References  to  Chapter  XV. : 

(1)  Cf.  Lord  Eayleigh,  Phil.  May.,  [5],  30,  285,  456,  33,  209. 

(2)  Lord  Kelvin,  Phil:  May.,  (iv.),  17,  61,  1859. 

(3)  J.  W.  Gibbs,  Scientific  Papers,  1. 

(4)  Frankenheiin,  Journ.  f.  prakt.  Chem.,  23,  401,  1841;  Lehre   ron  der 
Kohdsion,  86,  1836. 

(5)  E.  Eotvos,  Wied.  Ann.,  27,  452,  1886;  G.  A.  Einstein,  Drudes  Ann., 
4,  513,  1901. 

(6)  Eamsay  and  Shields,  Zeitschr.  physical.  Chem.,  12,  433,  1893  ;  Bamsay 
and  Aston,  ibid.,  15,   98,    1894;    Guye  and  Baud,   Hid.,   1(2,   379,    1903; 
cf.  Nernsfc,  Jahrb.  der  Chem.,  3,  18,  1893,  van  der  Waals.  Zeitschr.  physikal. 
Chtm.,  13,  713,  1894. 

(7)  E.  T.  Whittaker,  Proc.  Roy.  Soc.,  A,  81,  21,  1908;  E,  D.  Kleeman 
Phil.  Mag.,  [6],  11,  491,  901,  1909. 

(8)  McBain,  Trans.  Chem.  Soc.,  91,  1683,  1907. 

(9)  Faraday,  Phil.  Trans.,  1^,  55,  1834. 

(10)  H.  Freundlich,  Zeitschr.  physikal.   Chtm.,   57,   385.    1906;  cf.  also 
Kapillarchemie ;  see  also  the  author's  Higher  Mathematics,  §  91. 

(11)  J.  J.  Thomson,  Applications  of  Dynamics  to  Physics  and  Chemistry, 
London,  1888,  p.  191.     Cf.  Freundlich  and  Emslander,  Ztitschr.  physikal. 
Chem.,  49,  317,  1904;  Warburg,  Weid.  Ann.,  1^1,  14,  1890. 

(12)  Milner,  Phil.  Mag.,  [6],  13,  96,  1907. 

(13)  Zawidski,  Zeitschr.  physikal.  Chem.,  35,  77,  1900;   '4!,  612,  1903. 

(14)  Miss  Benson,  Journ.  Phys.  Chtm.,  7,  532,  1903. 

(15).  "W.  C.  McC.  Lewis,  Phil.  Mag.,  [6],  15,  498,  1908;  ibid.,  17,  466, 
1909. 

(16)  F.  G.  Donnan  and  J.  T.  Barker,  Proc.  Roy.  Soc.,  A,  85,  557,  1911. 

(17)  Duclaux,  Ann.  Chem.  phys.,  [5],  13,  76,  1878;  M.  Iraube,  Berl.  Ber., 
17,  2294,  1884 ;  J.  prakt.  Chim.,  3%,  292,  515,  1886 ;  Lieb.  Ann.,  265,  27, 1891 ; 
Forch,  Wied.  Ann.,  68,  801,  1899  ;  Whatmough,  Zeitschr.  physikal.  Chem., 
39,  129,  1902;  Drucker,  ibid.,  52,  641,   1905;  Eitzel,  Hid.,  60,  319,  1907; 
Linebarger,  Amer.   Journ.  Sci.,  [4],  2,   226,  1896;  Sutherland,  Phil.  Mag., 
[5],  38,  194  ;  Herzen,  Arch,  de  Sci.  phys.  et  nat.,  [4],  lit,  232, 1902  ;  Eontgen 
and  Schneider,  Wied.  Ann.,  29,  165,  1886. 

(18)  Valson,  Ann.  Chem.  Phys.,  [4],  20,  361,  1870. 

(19)  Szyszkowski,  Zeitschr.  physikal.  Chtm.,  6$,  385,  1908. 

(20)  Chappuis,  Wied.  Ann.,  19,  29,  1883. 

(21)  T.  de  Saussure,   Gilb.  Ann.,  £7,   113,   1814.     For  earlier  literature 
cf.  Ostwald,  Lehrbuch,  1,  1084,  2nd.  edit.  1891. 

(22)  Baerwald,  Drude's  Ann.,  23,  84,  1907. 


CAPILLARITY  AND  ADSORPTION  449 

(23)  M.  Travers,  Proc.  Boy.  Soc.,  A,  78, 9, 1906  ;  cf .  Miss  Homfray,  Ztitschr. 
physikaL  Chem.,  1\,  129,  1910. 

(24)  Favre,  Ann.  CJiim.  Phys.,  [5],  7,  209,  1874. 

(25)  Dewar,  Proc.  Roy.  Soc.,  7%,  122,  127,  1904  (low  temperatures). 

(26)  Parks,  Phil.  Mag.,  [6],  It,  240,  1902. 

(27)  Jungh,  Pogg.  Ann.,  1*5,  292,  1865. 

(28)  Schwalbe,  Drude's  Ann.,  16,  32,  1905. 

(29)  Freundlich,  KapiUarthemie,  115. 

(30)  Pawlow,  Zfitsrhr.  physikal.  Chem.,  75,  48,  1911. 

(31)  H.  Rodewald,  Zeitschr.  i.hysikaL   Chem.,  2$,  193,  1897  ;  cf.  Eeinke, 
Hanstens  lotan.  AbhandL,    'j,   1,   1879  ;  Wiedmann  and  Liideking,   ITiW. 
Ann.,  S.J,  145,  1885  ;  Pascheles,  Pfliiger's  Archives,  07,  225,  1897. 


T.  G    G 


CHAPTEK  XVI. 

ELECTROCHEMISTRY 

197.    The  Thermoelectric  Circuit. 

If  the  two  junctions  of  a  circuit  of  two  wires  of  different  metals 
are  maintained  at  different  temperatures,  TI  >  T2,  an  electric 
current  flows  round  the  circuit,  its  direction  and  magnitude 
depending  on  the  nature  of  the  metals  and  on  the  temperatures 
(Seebeck,  1821). 

If  TI  —  T2  is  small,  the  electromotive  force  acting  round  the 
circuit  is  approximately  proportional  to  it. 

The  source  of  the  energy  of  thermoelectric  currents  is  indicated 
by  the  observation  of  Peltier  (1834)  that  heat  is  absorbed  at  the 

hot  junction  and  evolved  at  the 
cold  junction,  and  that  if  the 
direction  of  the  current  is  re- 
versed by  inserting  a  battery  in 
the  circuit,  these  thermal  effects 
at  the  junctions  are  also  re- 
versed. The  heat  liberated  or 
absorbed  is  proportional  to  the 
quantity  of  electricity  crossing  the  junction,  and  for  unit  quantity 
is  denned  as  the  Peltier  effect,  TT  at  the  junction. 

Besides  the  reversible  production  of  heat  at  the  junctions,  there 
is  an  evolution  of  heat  all  round  the  circuit  due  to  frictional 
resistance,  this  Joule's  heat  being  proportional  to  the  square  of 
the  current,  and  hence  not  reversed  with  the  latter.  There  is 
also  a  passage  of  heat  by  conduction  from  the  hotter  to  the  colder 
parts.  But  if  the  current  strength  is  reduced,  the  Joule's  heat, 
being  proportional  to  its  square,  becomes  less  and  less  in  com- 
parison with  the  Peltier  heat,  and  with  very  small  currents  is 
negligible.  We  shall  further  assume  that  the  reversible  thermo- 
electric phenomena  proceed  independently  of  the  heat  conduction, 
so  that  the  whole  circuit  may  be  treated  as  a  reversible  heat 


ELECTROCHEMISTRY  451 

engine  (Lord  Kelvin  (1854),  the  junctions  being  placed  in  the 
two  heat  reservoirs  (Fig.  86). 

We  first  assume  that  the  Peltier  effects  are  the  only  reversible 
heat  effects  in.  the  circuit.  Then  if  TTI,  7r2  are  the  Peltier  effects 
at  the  hot  and  cold  junctions : 

TTi-TTa^E  .  .  .  .       (1) 

the  electromotive  force  acting  round  the  circuit,  i.e.,  the  work 
done  per  unit  quantity  of  electricity  circulating.  From  the 
entropy  equation : 

771  —  ^  f'->\ 

E~E  ' 

.•.ErrCTx-Ta)-^  ....     (3) 

so  that  with  a  fixed  temperature  of  the  cold  junction  the  electro- 
motive force  should  be  proportional  to  the  difference  of  the 
temperatures  of  the  junctions  (Clausius,  1853). 

It  was  known  even  to  Peltier  that  this  result  is  not  general ; 
there  are  circuits  the  electromotive  force  of  which  decreases  with 
rise  of  temperature  until  at  a  definite  so-called  inversion  tempera- 
ture, it  vanishes.  With  further  rise  of  temperature  the  electro- 
motive force  and  current  are  reversed.  The  existence  of  this 
thermoelectric  inversion,  or  Cummin g  effect,  led  Lord  Kelvin  to 
surmise  that  reversible  heat  effects  other  than  the  Peltier  effects 
existed,  and  he  succeeded  in  showing  experimentally  that  heat 
may  be  absorbed  or  evolved  when  a  current  flows  along  a  single 
homogeneous  wire  from  a  place  at  higher  to  one  at  lower  tem- 
perature, and  the  effect  is  reversed  with  the  current.  In  the 
case  of  copper  heat  is  evolved  when  the  current  flows  from  a  hot 
to  a  cold  portion,  in  the  case  of  iron  it  is  absorbed,  in  the  case  of 
lead  there  is  no  change.  This  is  referred  to  as  the  Thomson  effect. 
The  current  appears  to  convey  heat  convectively  from  one  part  of 
the  circuit  to  another,  much  as  a  current  of  water  in  an  unequally 
heated  tube. 

Let  crcW  be  the  heat  developed  per  second  in  a  portion  of  a 
homogeneous  conductor  the  ends  of  which  are  at  temperatures  8 
and  6  +  dO,  when  unit  current  passes  from  the  warmer  to  the 
colder  end.  cr  is  called  the  specific  heat  of  electricity  in  the  metal. 
Let  the  values  of  cr  in  the  arcs  (1)  and  (2)  be  cr1?  o-2  respectively. 

If  TO  is  the  temperature  of  a  chosen  position  in  the  arc  (1),  the 

G  G  2 


452  THERMODYNAMICS 

total  heat  developed  per  second  by  unit  current  all  round  the 
circuit  in  this  way  is  : 


p  p  p 

MT  +       <72r7T  +       ov/T  = 
JTI  Ji2  JTO 


-     <T2)    rfT   .      (1), 

whilst  the  quantity  absorbed  at  the  junctions  is 

TTi  —  7T2      .  .  .  .     '         .       (2) 

Let  E  be  the  electromotive  force  of  the  circuit,  i.e.,  the  energy 
developed  round  the  circuit  by  unit  current  per  second,  then 
from  the  First  Law : 

/*T2 
E  =  TTi  —  7T2  —       (fl-i  —  0-2)  f/T      .  .  .       (3) 


The  Entropy  Principle  gives,  on  the  assumptions  as  to  reversi- 
bility of  thermoelectric  effects  : 

T2 

1  ~  ^  fzTr=o  .     .     .  (4) 
T1  T 

If  T!  =  T2,  TTi  =  7T2,  then  E  —  0. 

If  TI  —  T2  is  infinitely  small,  i.e.,  a  very  weak  current  is  pass- 
ing, we  have,  by  differentiating  (4)  : 

o-i  —  0-2  _„  . 

~" 


77          f/7T 
—  0-2  =    rp  —  ^      •  •  •  •       (6) 


r 

=- 

V   r 


^E   ....     (7) 

Ti 

=  ^  (Ti  -  T2)  if  T!  -  T2  is  small. 
By  differentiating  (7)  we  obtain  the  Peltier  effect  at  a  junction : 


The  Peltier  effect  at  a  single  junction  is  therefore  equal  to  the 
absolute  temperature  of  the  junction  multiplied  by  the  rate  of 


ELECTROCHEMISTRY  453 

decrease  with  temperature  of  the  total  electromotive  force  round 
the  circuit. 

For  H  different  metals  in  a  circuit  : 


and 


are  the  equations  from  which  the  thermoelectric  effects  in  the 
circuit  may  be  derived. 

198.     Theories  of  Thermoelectricity. 

The  various  theories  which  have  been  proposed  to  account  for 
the  phenomena  of  the  thermoelectric  circuit  may  be  grouped  into 
three  classes  : 

(1)  The  theory,  due  to  Clausius,  that  in  the  thermoelectric 
current,  as  in  other  electric  currents,  the  electricity  alone  moves, 
under  the  impulse  of  the  constantly  maintained  potential  differ- 
ences at  the  junctions. 

(2)  The  theory  of  F.  Kohlrausch  (1875),  according  to  which 
every  electric  current  is  intimately  associated  with  a  thermal 
current,  and  vice  versa,  and  heat  is  conveyed  by  an  electric  current. 
The  source  of  the  thermoelectric  energy  is  then  located  in  the 
interior  of  the  unequally  heated  homogeneous  wires,  and  the 
junctions  have,  according  to  this  theory,  only  a  secondary  influ- 
ence.    Kohlrausch  cites  as  evidence  for  the  intimate  connexion 
between  the  two  currents  the  proportionality  which  exists  between 
the  conductivities  of  a  metal  for  heat  and  for  electricity  (law  of 
Wiedemann  and  Franz). 

(3)  The  theory  of  Kelvin  (1854),  developed  in  the  preceding 
section,  stands  midway  between  these  two  hypotheses,  in  that  it 
assumes  the  existence  of  potential  differences  at  the  junctions, 
playing  the  role  postulated    by  Clausius,  and  also  admits  the 
production  of  electromotive  forces  in  the  interior  of  the  homo- 
geneous wires  due  to  inequalities  of  temperature  in  the  latter, 
these  inequalities  giving  rise  to  the  flow  of  heat  which  is  regarded 
as  essential  in  the  theory  of  Kohlrausch. 


454  THERMODYNAMICS 

Clausius  saw  as  the  origin  of  the  electromotive  forces  at  the 
junctions  the  formation  of  a  so-called  double  layer,  consisting  of 
a  sheet  of  positive  electrification  on  the  surface  of  one  metal 
and  a  sheet  of  negative  electrification  on  the  surface  of  the 
other  metal.  This  double  layer  (I)oppehclticlit),  according  to 
Helmholtz  (1879)  arises  from  the  unequal  affinities  of  different 
metals  for  electricity.  There  is  some  cause — called  by  Planck 
(1889)  an  elect  romolecular  force — which  keeps  this  double  layer 
constantly  renewed.  Clausius  also  assumed  that  by  the  action 
of  heat  this  double  layer  was  disturbed,  and  electricity  driven 
from  one  metal  to  the  other,  and  the  work  done  by  the  absorbed 
heat  goes  to  produce  this  motion  of  electricity  against  the  electro- 
motive force  of  the  double  layer.  This  theory,  as  we  saw,  is  too 
simple,  and  leads  to  consequences  in  opposition  to  experience. 
Clausius,  and  Budde  (1884),  then  broadened  the  basis  of  the 
theory  by  assuming  that  differences  of  potential  also  arise  in  the 
single  wires,  but  this  is  nothing  more  than  the  theory  developed 
by  Kelvin  in  1854. 

Kohlrausch's  theory  leaves  quite  unexplained  the  fact  that  no 
thermoelectric  current  is  set  up  in  a  homogeneous  wire  along 
which  a  current  of  heat  is  flowing,  whilst  the  theory  of  Lord 
Kelvin  is  difficult  to  reconcile  with  the  fact  that  thermoelectric 
currents  cannot  be  set  up  in  a  circuit  of  liquid  metals,  although 
these  show  the  Thomson  effect.  The  latter  seems,  therefore,  to 
be  to  a  certain  extent  independent  of  the  Peltier  effect.  Theories 
intended  to  escape  these  difficulties  have  been  proposed  by  Planck 
(1889),  and  Duhem,  in  which  the  conception  of  the  entropy  of 
electricity  is  introduced. 

With  the  revival  of  the  corpuscular  theory  of  electricity  came 
also  new  theories  of  the  origin  of  thermoelectric  phenomena,  and 
the  starting-point  of  these  is  based  on  Kohlrausch's  theory. 
Eiecke  (1898)  assumed  the  existence  of  positive  and  negative 
particles,  in  part  bound  to  the  metallic  molecules,  in  part  free. 
The  motion  of  these  particles,  caused  by  an  electrical  potential 
difference  or  an  inequality  of  temperature,  gives  rise  to  a 
current  of  electricity,  and,  since  they  possess  kinetic  energy,  to  a 
current  of  heat.  The  theory  of  Drude  (1900)  is  the  most  com- 
plete attempt  to  refer  the  thermoelectric  phenomena  to  the 
properties  of  electrons.  The  free  positive  and  negative  electrons 
(Kern en}  are  assumed  to  move  amongst  the  metallic  atoms, 


ELECTROCHEMISTRY  455 

and  it  is  further  supposed  that,  like  the  molecules  of  a  gas,  they 
have  a  kinetic  energy  proportional  to  the  absolute  temperature. 
A  motion  of  these  charges  will  include  a  transport  of  kinetic 
energy,  which  corresponds  with  the  flow  of  heat,  and  a  transport 
of  electricity,  which  is  the  electric  current.  From  his  theory, 
Drude  was  able  to  calculate  the  magnitude  of  the  charge  of 
an  electron,  and  the  number  agrees  with  that  obtained  by 
•T.  J.  Thomson. 

The  negative  charges  are  free  electrons,  the  positive  charges 
appear  to  be  largely  bound  to  metallic  atoms  as  positive  ions. 

(E.  Riecke,  Ann.  Phys.,  66,  353,  545,  1199,  1898;  Zeitschr. 
Eltktrochem.,  1909;  P.  Drude,  ibid.  [4],  1,  566;  3,369,1900; 
H.  A.  Lorentz,  Archives  Xeerland.  II.,  10,  1905,  assumes  that 
only  negative  electricity  is  free  in  the  metal,  the  positive  existing 
as  metal  ions). 

199.    Voltaic  Cells. 

A  voltaic  cell  is  an  arrangement  by  which  an  electric  current 
is  obtained  through  the  agency  of  material  changes  in  the  con- 
stituents of  the  cell. 

This  definition  excludes  such  sources  of  current  as  thermo- 
couples, or  dynamos,  in  which  all  the  materials  composing  the 
element  remain  unchanged. 

A  voltaic  cell  consists  essentially  of  three  parts  :  two  electrodes, 
from  which  the  positive  and  negative  electricity  leave  the  cell, 
and  an  electrolyte  in  which  the  electrodes  are  contained.  Its  form 
is  therefore  that  of  an  electrolytic  cell,  and  the  difference  between 
the  two  lies  only  in  the  condition  that  in  the  former  we  produce 
an  electric  current  through  the  agency  of  the  material  changes, 
whereas  in  the  latter  we  induce  these  material  changes  by  a 
current  supplied  from  an  external  source :  the  same  arrangement 
may  therefore  serve  as  either.  The  direction  in  which  the 
current  flows  through  the  cell  will  depend  on  the  potential 
difference  between  its  terminals. 

The  electromotive  force  of  a  voltaic  cell  is  the  total  amount  of 
work  done  when  unit  quantity  of  electricity  passes  through  the 
cell. 

When  the  cell  is  in  action,  a  definite  chemical  reaction  occurs 
in  its  interior,  and  according  to  Faraday's  laws  the  amount 
of  chemical  decomposition  is  proportional  to  the  quantity  of 


456  THERMODYNAMICS 

electricity  passing,  and  to  the  chemical  equivalent  of  the  ion 
transported.  For  every  chemical  equivalent  of  ion,  96540 
coulombs  of  electricity,  positive  or  negative  according  as  the  ion 
is  a  kation  or  an  anion,  pass  round  the  circuit.  This  quantity  is 
called  a  faraday,  F. 

A  voltaic  cell  is  said  to  be  reversible  when  an  opposing  electro- 
motive force  greater  by  an  infinitesimal  amount  than  that  of  the 
cell  reverses  the  direction  of  the  current,  and  the  material  changes 
occurring  in  the  cell. 

An  example  of  a  reversible  cell  is  the  Weston  normal  element : 

Hg  /  Hg2S04  /  CdS04  aq.  /  Hg  +  Cd. 
the  reaction  in  which  : 

Cd  +  Hg2S04  =2  2Hg  +  OdS04 

may  proceed  readily  in  either  direction  according  as  the  counter 
electromotive  force  is  slightly  less,  or  slightly  greater,  than  that 
of  the  cell,  1-0187— 0'35  X  l()-*(0  —  18)  volt  at  0°C. 

Such  a  reversible  cell  may  be  used  as  a  source  of  external  work 
by  attaching  its  poles  to  the  two  plates  of  an  electrostatic  air- 
condenser,  or  to  an  ideal  electromotor.  Slight  motions  of  the 
parts  of  these  systems  in  one  direction  or  the  other  against  the 
external  forces  holding  them  in  equilibrium,  will  yield  or  absorb 
external  work,  and  cause  corresponding  currents  to  pass  round 
the  system. 

200.    The  Gibbs-Helmholtz  Equation. 

There  is  a  very  important  equation  relating  to  the  electromotive 
forces  of  reversible  cells  which  was  deduced  independently  by 
J.  Willard  Gibbs  (1875)  and  H.  von  Helmholtz  (1882),  and  is 
usually  called  the  Gibbs-Helmholtz  Equation. 

If  a  reversible  cell  is  connected  with  an  external  balancing 
electromotive  force  capable  of  slight  variation,  we  can  allow  a 
quantity  of  electricity  F  =  96540  cmb.  to  flow  through  the  cell 
in  the  current-producing  direction  whilst  the  whole  is  maintained 
at  a  constant  temperature  T. 

The  material  changes  in  the  cell  are  completely  defined  when 
we  know  the  quantity  of  electricity  passing  through,  for  Faraday's 
law  teaches  us  that  for  a  quantity  F  there  will  always  be  a  gram- 
equivalent  of  chemical  change,  independent  of  the  electromotive 
force. 


ELECTEOCHEMISTRY  457 

From  the  definition  of  AU  we  have  : 

AU  =  2Q-2A      .         .         .         .     (1) 

\vhere  2Q  is  the  heat  absorbed  from  the  constant  temperature 
reservoir,  2A  is  the  external  work  done,  and  AU  is  the  total 
increase  of  the  intrinsic  energy  of  the  cell. 

If  A  is  the  amount  of  heat  absorbed  per  F,  from  the  reservoir 
at  the  temperature  T  :  - 

2Q  =  A     .....     (2) 

is  called  the  latent  heat  of  the  cell. 

If  the  changes  of  volume  occurring  as  a  result  of  the  chemical 
reaction  are  negligibly  small,  as  when  liquids  or  solids  alone 
participate  in  it,  the  external  work  is  wholly  electrical  : 

2A  =  EF  .         .         .         .     (3) 

Now  if  the  chemical  reaction  had  been  allowed  to  proceed 
without  the  performance  of  any  external  electrical  work,  say  in 
a  calorimeter,  so  that  the  initial  and  final  temperatures  of  the 
system  are  both  T,  the  change  of  intrinsic  energy  would  have 
been  the  same  as  that  occurring  in  the  process  described  above, 
as  we  know  from  the  First  Law.  Thus  the  heat  of  reaction,  Q 
will  be  equal  to  the  increase  of  intrinsic  energy  : 

AU  =  Q  .....     (4) 

From  (1),  (2),  (3),  and  (4)  we  obtain  : 

£  =  A-EF         .         .         .         .     (5) 

which  is  the  equation  expressing  the  First  Law  for  galvanic  cells 
in  -which  the  volume  changes  are  small. 

Since  the  process  is  isothermal  and  reversible,  we  may  apply 
the  equation  of  §  58  as  a  special  case  of  the  Second  Law  : 

U  =  T-.         .         .         •     (6) 


in  which  the  partial  differential  coefficient  refers  to  a  constant 
amplitude,  i.e.,  to  a  constant  quantity  of  electricity  F  transported, 
and  the  thereby  defined  chemical  reaction. 
Now  AU  =  Q  =  A  -  EF 

AT  =  EF 

and  = 


458  THEKMODYNAMICS 

thence :  A  =  FT  ^         .      "  .   -     .         .     (7) 

The  differential  coefficient  is  total,  since  E  does  not  depend  on 
the  extent  of  the  chemical  change. 

Thus,  if  we  find  how  the  electromotive  force  changes  when  the 
temperature  of  the  cell  is  altered  on  "  open  circuit,"  i.e.,  when  no 
current  is  passing,  we  can  at  once  calculate  A,  the  latent  heat, 
just  as  we  can  calculate  the  latent  heat  of  evaporation  of  a  liquid 
when  we  know  the  variation  of  its  vapour  pressure  with  tempera- 
ture. Since  E  changes  only  slightly  with  T,  we  can  evaluate 

-TTp  by  the  method  of  mean  value  (H.M.,  §  69) : 

cm        T:  +  T2  _  E2  -  Ex 

<7fft1     — 2~     "fT^Ti     '         '         ' 

From  (5)  and  (7)  we  find : 

EF  +  Q  =  FT H   ....     (9) 

Let  Q/F  =  q,  the  heat  of  reaction  per  electrochemical  equivalent, 
then: 

E  +  q  =  T    ^     .  .  .  .      (10) 

Equations  (7),  (9),  and  (10)  are  different  forms  of  the  Gibbs- 
Helrnholtz  equation,  which  is  the  fundamental  equation  of 
electrochemistry. 

From  (9)  we  see  that  there  are  three  possibilities : 

7T? 

(i.)  if  ^m>0,  i.e.,  the  electromotive  force  increases  with  rue  of 

temperature,  then  EF  +  Q  >  0,  hence  A  >  0,  so  that  the  cell 
absorbs  heat  in  action  ; 

7TJ1 

(ii.)  if  y™  =  0,  i.e.,  the  electromotive  force  is  independent  of 

temperature,  then  EF  +  Q  —  0,  hence  A  =  0,  i.e.,  the  cell  neither 
absorbs  nor  emits  heat  in  action ; 

F 
(iii.)  if  -rp-  <  0,  i.e.,  the  electromotive  force  decreases  icith  rise 

of  temperature,  then  EF  +  Q  <  0,  hence  A  <  0,  so  that  the  cell 
emits  heat  in  action. 


ELECTROCHEMISTRY  459 

Cases  (i.)  and  (iii.)  indicate  that  there  is  a  change  of  entropy 
in  the  corresponding  processes,  since 

A  =  T(Sa  -  SO   .          .          .          .  (IQa) 

7TP 

.-.  if  -™-  >  0,  82  >  Si,  and  the  entropy  of  the  cell  increases, 
if  -7777-  <  0,  S2  <  Si,  and  the  entropy  of  the  cell  decreases. 

The  total  entropy  of  the  cell  and  heat  reservoir  remains  con- 
stunt,  since  the  process  is  reversible. 

If  ~  =  0  we  see  that  EF  =  -  Q,  so  that  the  electrical  work 

done  by  the  cell  is  exactly  equal  to  minus  the  heat  of  reaction. 
Lord  Kelvin  (1851),  who  found  this  relation  verified  in  the  case 
of  the  Daniell  cell,  assumed  that  it  held  generally,  i.e.,  the 
electrical  work  done  by  a  cell  is  equal  to  the  diminution  of  chemical 
energy  of  its  components. 

The  heat  of  reaction  when  one  electrochemical  equivalent  of 
zinc  displaces  copper  in  sulphate  solution  is  2*592  cal.  =  —  q 

.'.  E  =  2-592  X  4-18  X  107  =  1'09  X  108  E.M.  units. 
=  1'09  volt,  which  agrees  with  the  observed  value. 

As  Gibbs  and  Helmholtz  pointed  out,  however,  this  so-called 
Thomson  Rule  (which  had  also  been  proposed  by  Helmholtz  in 
1847)  cannot  be  true  in  general,  because  the  change  of  entropy 
cannot  always  be  neglected. 

Cases  in  which  a  cell  cools  or  warms  itself  in  action  had  been 
investigated  by  Brauri  (1878 — 1883),  and  the  quantitative  relation 
was  verified  in  a  number  of  cases  by  Jahn  (1886),  who  measured 
the  latent  heats  by  placing  the  cell  in  an  ice  calorimeter. 

In  the  Clark  cell  at  0°,  according  to  E.  Cohen  : 

Q  =  -  340730  j. 

2  EF  =  1-4291  volt  X  2  X  96540  cmb.  =  275930  j. 
.-.  A  =  —  64800.;',  i.e.,  the  cell  converts  only  76  per  cent,  of 
its  chemical  into  electrical  energy,  and  emits  the  rest  as  heat. 
The  temperature  coefficient  of  the  Daniell  cell  is  -f  0'000034 

,  its  E.M.F.  at  0°  C.  is  1*0962  volt 
degr. 

.-.  2EF  =  2  X  96540  X  1*0962  =  50526  j. 


460  THERMODYNAMICS 

The  heat  of  reaction  is  Q  =  —  50110.;. 

.*.  the  latent  heat  is  416  j. 

whereas   2FT  ^  =  2  X  96540  X  273  X  0'000034  =  428  j. 

The  direct  measurements  of  Jahn  (1888,  1893)  and  of  Gill 
(1890)  show  that  the  latent  heat  A  arises  at  the  surfaces  of  contact 
of  the  electrodes  and  electrolyte  and  is  fully  accounted  for  by 
these  Peltier  heats  at  the  junctions  of  conductors.  The  equation 
of  §  197  : 

*.=  -T||  -     ,  -   (ID 

where  £TT  is  the  sum  of  the  Peltier  effects  round  the  circuit, 
and  E  is  the  total  electromotive  force,  was  found  to  apply,  and 
hence  the  Gibbs-Helmholtz  equation  may  be  written  in  the  form  : 

E  +  q  =  —  £77 
or  q  =  —  (E  +  STT)  .          .         .     (12) 

The  heat  of  formation  of  a  substance  in  a  voltaic  cell  may 
therefore  be  calculated  from  the  measured  Peltier  effects  and  the 
electromotive  force. 

The  integral  of  the  Gibbs-Helmholtz  equation  is  : 


where  C  is  the  integration  constant,  since  the  differential  equation 
(10)  is  easily  transformed  into  : 

/EX 

d    W      • 

rfT      T2 

If  q  is  independent  of  T,  i.e.,  the  total  heat  capacity  of  the 
initial  reacting  substances  is  equal  to  that  of  the  products  of 
reaction  (§  58),  then: 

.      ..."     ~.     (14) 

If  there  exists  a  temperature  T0  at  which  the  electromotive 
force  of  the  cell  vanishes  : 

.         .         .     (15) 


ELECTROCHEMISTRY  461 

and  E  =  q0  T  ~  T°  ....     (16) 

an  equation  due  to  Gibbs  (1886). 

TO  is  called  the  transition  temperature,  and  at  this  temperature 
the  chemical  reaction  in  the  cell  would  go  on  reversibly  in  either 
direction,  since  its  progress  involves  no  change  of  available  energy. 
If  two  of  the  magnitudes  E,  T0,  T  are  known,  the  third  can  be 
calculated  from  a  knowledge  of  q.  Since  T0  can  be  determined 
by  non-electrical  methods  (e.g.,  by  measurements  of  solubilities,  or 
changes  of  volume,  etc.)  the  equation  serves  to  determine  the 
electromotive  force  of  a  voltaic  cell  without  actually  setting  up 
the  latter,  as  was  emphasised  by  Gibbs. 

The  method  has  been  applied  by  Cohen  (1894)  to  the  deter- 
mination of  transition  temperatures.  Thus  the  electromotive 
force  of  the  cell : 

Zn  |  sat.  sol.  ZnS04 .  7H20  |  sat.  sol.  ZnS04 .  6H20  |  Zn 
vanishes  at  the  transition  temperature  of  the  reaction  : 
ZnS04 .  7H20  =  ZnS04 .  6H20  +  H20 

(cf.  §  11)  at  which  the  two  hydrates  have  the  same  solubility. 
If  the  temperature  is  raised  above  T0,  the  polarity  is  reversed. 

The  existence  of  a  transition  temperature  at  which  E  vanishes 
permits  of  a  very  important  transformation  of  the  equation  (13), 
viz.,  it  enables  us  to  replace  an  indefinite  integral  by  a  definite 
integral,  in  that  a  fixed  lower  limit  of  integration  may  be  assigned 
to  the  former.  The  equation  (13)  may  now  be  written  : 

T    .         .         .         .     (17) 

and  the  electromotive  force  is  therefore  defined  without  ambiguity 
in  terms  of  the  thermal  magnitudes.  We  shall  make  use  of  a 
similar  transformation  later  (§  209). 

201.     Effect  of  Pressure  on  the  Electromotive  Force. 

If  the  change  of  volume  occurring  in  the  cell  is  taken  into 
account,  which  is  particularly  of  importance  when  gases  partici- 
pate in  the  reaction,  we  may  proceed  as  follows : 

For  a  small  reversible  change : 

rfU  =  2SQ  -  2SA  =  TrfS  -  Ede  -  pdV .        .    (1) 


462  THERMODYNAMICS 

where,  in  addition  to  the  electrical  term  ~E>dc,  e  being  the  quantity 
of  electricity    flowing    round    the    circuit,    the    external   work 
involves  a  term  pdV  due  to  expansion  of  the  whole  system  by 
dV  under  an  external  (e.g.,  atmospheric)  pressure  p. 
Subtract  rf(ST  —  pV)  from  each  side  of  (1)  : 

.-.  d(U  -  TS  +  2>V)  =d<j>  =  -  SrfT  -  E<fo  +  \dp     .     (2) 
where  c/>  =  U  -  TS  +  pV 

is  the  thermodynamic  potential  of  the  system. 
Since  d(f>  is  a  perfect  differential  we  have  : 

^=-E,^  =  V,|=-S      .        .         .    (8) 

de  op  9T 

and  hence  such  relations  as  : 

8E\  .       a  /\  _       a  /a\  _       /av 


a7  V^    -  f  ' 

i.e.,  the  electromotive  force  increases  with  the  pressure  at  a  rate 
equal  to  the  rate  of  decrease  of  the  total  volume  at  constant 
pressure  per  unit  quantity  of  electricity  passing  round  the  circuit, 
the  temperature  in  both  cases  being  constant.  According  to 
Faraday's  law  the  change  of  volume  depends,  at  a  given  tempera- 
ture and  pressure,  only  on  the  quantity  of  electricity  passing  : 


where   YO,   Y    are   the    volumes   before   and   after   the    electric 
transfer.     For  chemically  equivalent  amounts  e  =  F. 

Thus  jjy  =  V°  ~  V        .         .         .         .     (6) 

If  only  condensed  phases  are  present,  YO,  Y  are  practically 
independent  of  pressure,  hence  at  a  constant  temperature  the 
integral  of  (6)  is,  for  this  case : 

^-El  =  ^^(p2-Pl)    .        .        .     (7) 

which  has  been  verified  by  Cohen  for  the  Clark  cell,  and  also  by 
other  investigators  for  other  cells. 

If  gases  participate  in  the  reaction,  we  can  put : 

:  (8) 


ELECTROCHEMISTRY  463 

where  A  refers  to  the  condensed  part,  which  suffers  a  change  of 
volume  practically  independent  of  temperature  and  pressure,  and 

—  refers  to  the  volume  of  the  gas  phase,  which  is  proportional 

to  the  absolute  temperature  and  inversely  proportional  to  the 
pressure.     The  interpretation  of  A  and  B  is  simple. 
Thus,  by  integration  of  (6) : 

E2  —  EX  =  A  (#,  —  2>i)  +  BT/H  y        •         •     (9) 

at  constant  temperature. 

This  equation  has  been  verified  by  Gilbaut  (1891). 

202.     Concentration  Cells. 

In  cells  like  the  Daniell,  the  electrical  energy  is  derived  from 
chemical  changes  occurring  between  the  electrodes  and  electro- 
lytes ;  the  final  system  possesses  less  free  chemical  energy  than 
the  initial,  and  the  difference  goes  over  into  free  electrical  energy. 
A  voltaic  cell  may  also  derive  its  energy  not  from  changes  of 
composition,  but  from  changes  of  concentration,  which  may  occur 
in  the  electrodes  or  in  the  electrolyte. 

If  we  assume  that  the  heat  of  dilution  is  zero,  which  limits  the 
discussion  to  dilute  solutions  : 

ff  =  0 (1) 

.'.  from  the  Gibbs-Hehnholtz  equation  : 

E  =  T—  or  ^  =  ^ ; 

.-.  /wE  =  InT  +  const. 

/.  E  =  CT (2) 

The  total  inapplicability  of  the  Thomson  rule  to  this  case  is 
at  once  apparent;  none  of  the  electrical  energy  comes  from 
chemical  change,  but  the  cell  functions  as  a  heat  engine, 
converting  the  heat  of  its  environment  into  electrical  work. 

The  theory  of  concentration  cells  was  first  developed  with  great 
generality  by  Helmholtz  (1878),  who  showed  how  the  electro- 
motive force  could  be  calculated  from  the  vapour  pressures  of  the 
solutions,  and  his  calculations  were  confirmed  by  the  experiments 
of  Moser  (1878). 

The  simplest  case  is  one  hi   which  we  have  two   reversible 


464  THERMODYNAMICS 

electrodes  in  a  gas  at  different  pressures,  for  example,  two 
platinum  plates  saturated  with  hydrogen,  dipping  into  acidulated 
water,  and  surrounded  by  hydrogen  gas  at  the  pressures  pi,  p%. 
If  these  are  connected,  gas  dissolves  at  the  higher,  and  is  evolved 
at  the  lower,  pressure.  Let  us  suppose  the  cell  works  against  a 
balancing  electromotive  force  until  1  mol  of  gas  has  passed  from 
the  higher  to  the  lower  pressure.  The  electrical  work  is  : 

E  X  2F 
since  each  molecule  of  hydrogen  yields  two  ions. 

If  we  remove  the  niol  of  gas  from  the  low  pressure  space, 
compress  it  isothermally  and 'reversibly  until  its  pressure  rises  to 
p%,  and  then  introduce  it  into  the  high  pressure  space,  the  cycle 
will  be  completed.  The  work  done  with  the  gas  is : 


/•P2  /V>2 

—    I  pdv  —  2W%  =    \  Vt 
JPI  J« 

^le's  law,  j 

r         / 
1    ~     J 


vdp 
If  the  gas  obeys  Boyle's  law,  p\r\  = 


pdr  =  -  RT  In      . 

pi 
I 

This  must  be  equal  to  the  electrical  work,  since 


No  experiments  appear  to  have  been  made  with  such  cells, 
although  the  equation  has  been  verified  with  oxygen  at  different 
partial  pressures  in  admixture  with  nitrogen,  with  platinum 
electrodes  and  hot  solid  glass  as  electrolyte  (Haber  and  Moser). 
A  similar  case  is  that  of  two  amalgams  of  a  metal,  of  different 
concentrations,  as  electrodes,  and  a  solution  of  a  salt  of  the  metal 
as  electrolyte  (G.  Meyer,  1891).  Here  we  must  take  the  osmotic 
pressures  of  the  metals  in  the  amalgams,  PI,  P2,  and,  for  an 
?i-valent  metal  : 


But 

,.E  =         te          ....     (5) 


ELECTROCHEMISTRY 


465 


It  has  been  found,  however,  that  this  case  is  somewhat  compli- 
cated by  the  formation  of  definite  compounds  in  some  amalgams ; 
still  the  general  results  are  in  agreement  with  the  theory.  Some 
exceptional  cases  found  by  Meyer  have  recently  been  shown  to 
depend  on  the  large  heats  of  dilution  of  the  particular  amalgams 
(Smith,  Zeitsclu:  anorg.  Chan.,  58,  381). 

The  equation  shows  that  the  electromotive  force  is  proportional 
to  the  absolute  temperature,  which  was  verified  by  experiment. 
It  is  also  independent  of  the  nature  of  the  electrolyte. 

A  very  important  practical  case  of  concentration  cell  is  that  in 
which   two  electrodes  of  the  same   material   are   immersed  in 
solutions  of  an   electrolyte  of  different 
concentrations.      Thus,    if    two    silver    Ay, 
plates    are    immersed   in    solutions    of 
silver  nitrate  of  different  concentrations, 
and  are  connected  by  a  wire,  the  metal 
dissolves  in  the  dilute  solution  and  is 
precipitated  from  the  strong  solution,  and 
this  goes  on  until  both  solutions  have 
the  same  concentration.    Let  us  consider 
a  cell  containing  the  solutions  of  con- 
centration £1,  £2  in  two  chambers  A  and 
B,  separated  by  a  porous  partition,  and 

containing  silver  plates.  When  F  coulombs  pass  round  the 
circuit  the  following  changes  occur,  where  n  is  the  migration 
ratio  of  the  anion  : 


ir 


(l-n) 


NO, 


(1)  Vessel  A. 

Gains  1  equiv.  Ag  by  dissolu- 
tion from  the  electrode. 

Loses  (1  —  n)  equiv.  Ag  by 
migration. 

Gains  n  equiv.  N03  by  migra- 
tion. 

/.  gains  n  equiv.  AgN03. 


(2)  Vessel  B. 

Gains  (1  —  »)  equiv.  Ag  by 
migration. 

Loses  1  equiv.  Ag  by  deposition. 

Loses  n  equiv.  N03  by  migra- 
tion. 

.'.  loses  n  equiv.  AgN03. 


The  nett  result  is  a  transference  of  n  equiv.  AgN03  from  the 
strong  solution  to  the  weak  solution,  where  n  is  the  migration 
ratio  of  the  anion. 

The  cycle  may  be  completed  by  returning  the  n  equiv.  of 
salt  to  the  weak  solution  by  a  reversible  osmotic  process.  Place 

T.  H    H 


466  THERMODYNAMICS 

the  strong  solution  in  a  cylinder  under  a  semipermeable  piston 
covered  with  pure  water,  and  allow  solvent  to  enter  reversibly 
until  the  concentration  of  the  solution  is  restored  to  its  value  (&) 
before  the  process.  If  the  volume  of  solution  is  large,  the  osmotic 
pressure  remains  practically  constant,  and  the  work  done  is  P^i, 
where  i\  =•  change  of  volume.  Now  separate  a  portion  of  solution 
containing  n  mols  of  AgN03,  and  compress  it  reversibly  until  its 
osmotic  pressure  rises  to  P2,  corresponding  with  the  concentration 

fP2 
PJr.     This 

solution  is  now  mixel  with  the  identical  strong  solution,  and 
solvent  is  removed  by  the  semipermeable  piston  till  it  regains  its 
original  volume. 

The  work  done  is  —  Pat'2. 

The  cycle  is  isothermal  and  reversible  : 

.-.     2  A  =  EF  +  fpi-l  *—     Pile  =  0 
L     Js    Jft 

/•Bs 

or  EF  =  —     vdP     .         .         .         .     (6) 

JPl 

We  shall  assume  that  the  solutions  are  so  dilute  that  the 
electrolyte  can  be  regarded  as  completely  ionised.  Then,  for  a 
binary  electrolyte,  for  the  amount  considered  : 

Pi;  =  2/iRT          .         .         .         .     (a) 


if  we  regard  n  as  the  limiting  value  of  the  transport  number 
attained  at  infinite  dilution,  or  since  P2/Pi  =  ?i/r2  =  £2/£i : 

& 

-g     •         •         •         •     (7) 

£1,  £2  refer  to  the  concentrations  round  the  kathode  and  anode, 
respectively,  and  the  negative  sign  shows  that  the  potential  of 
the  electrode  in  the  stronger  solution  is  less  than  that  in  the 
weaker  solution. 


ELECTROCHEMISTKY  467 

If  the  solutions  are  more  concentrated,  this  formula  does  not 
apply,  for  two  reasons  : 

(i.)  The  transport  number  n  is  a  function  of  the  concentration  ; 
(ii.)  The  equation  (a)  is  no  longer  valid. 
In  this  case  Lehfeldt  writes  : 

Pr  =  niET 

where  /  is  simply  the  ratio  between  the  actual  osmotic  pressure 
and  that  calculated  from  (a). 
Then 

....     (8) 


which  may  be  regarded  as  giving  the  osmotic  pressure  P  of  a 
solution  of  moderate  concentration. 
If  we  assume,  with  Arrhenius,  that  : 


where  ft  is  the  number  of  ions  produced  from  one  molecule  of 
electrolyte,  and  take  n  as  denoting  a  mean  value  between  those 
for  the  two  solutions : 

E_  _  RT»  " 


Now,  according  to  Ostwald's  dilution  law : 

a2          v    _  K  _  K  (1  +  a)/*RT 
l^a~         "I"          ~^~ 
.'.      /»P  =  /»(!  +  a)  +  //i(l  —  a)  -  'Una  — 
.-.  dln¥=dln(l  +  a)  +  dln(l  —  a)  —  -Mlna(T  const.) 

»2 


(1  +  a)  [tottna  -  tUn(l  +  a)  -  cU»(l-a)] 
L  J 


an  equation  due  to  Xernst  (1892).  This  is  of  very  wide  applica- 
bility, for  by  its  help  we  can  determine  either  ti,  or  £,  or  a,  for  a 
dilute  solution. 

H  H  2 


468  THERMODYNAMICS 

The  lead  accumulator  has  been  studied  from  the  thermo- 
dynamic  point  of  view  by  Dolezalek.  The  reaction  is : 

Pb02  +  Pb  +  2H2S04:T2PbS04  +  2H20 

If  two  such  accumulators,  containing  differently  concentrated 
acids,  are  connected  in  opposition,  i.e.,  both  peroxide  plates 
together  and  both  lead  plates  together,  the  one  with  the  stronger 
acid  is  discharged  in  the  sense  of  the  above  equation  from 
left  to  right,  whilst  the  one  with  the  weaker  acid  is  charged  in 
the  sense  of  the  equation  from  right  to  left.  The  reaction 
involves  the  passage  of  2F,  as  we  see  on  writing  down  its  anodic 
and  kathodic  components  : 

Pb  +  S04  =  PbS04 +2  0  (anode) 

Pb02  +  H2S04  +  2H  +  2  9  =  PbS04  +  2H20  (kathode) 

The  quantities  of  Pb,  Pb02,  PbS04  remain  unaltered,  provided 
the  substances  produced  are  in  the  same  state  of  aggregation,  etc., 
as  those  disappearing,  and  the  nett  result  is  the  transport  (of 
course,  apparent,  since  there  is  no  material  connexion  between 
the  cells)  of  2  mols  of  acid  from  the  stronger  (I.)  to  the 
weaker  (II.)  solution,  and  2  mols  of  water  in  the  opposite 
direction.  The  weak  acid  is  concentrated  and  the  strong  acid  is 
diluted,  the  process  taking  place  in  such  a  direction  as  to  equalise 
the  concentrations  as  if  the  acids  had  been  directly  mixed,  and 
the  electrical  work  produced  is  equal  to  the  free  energy  which 
would  have  been  lost  in  the  spontaneous  intermingling  of  the 
two  liquids.  This  free  energy  can  be  calculated  if  we  can 
construct  some  process  in  which  the  mixing  can  be  performed 
isothermally  and  reversibly,  since  it  is  equal  to  the  maximum 
work  of  this  process.  The  changes  of  concentration  may  be 
very  simply  brought  about  by  Helniholtz's  method  of  isothermal 
distillation. 

Solution  I.  has  lost  2H2S04and  gained  2H20,  whilst  solution  II. 
has  gained  2H2S04  and  lost  2H20.  This  change  may  also  be 
carried  out  as  follows  : 

(i.)  Take  from  I.  an  amount  of  solution  containing  2H2S04, 
say  2(H2S04  +  »iH20),  and  distil  water  from  it  to  the  solution  L, 
over  which  its  vapour  pressure  is  p±,  until  (almost)  pure  acid 
remains.  [The  removal  of  the  very  last  trace  of  water  would, 


ELECTROCHEMISTRY  469 

from  the  equation,  require  an  infinitely  large  expenditure  of  work, 
but  since  the  amount  remaining  in  the  acid  can  be  made  as  small 
as  we  please,  the  integral  has  a  finite  limiting  value."  The  work 
done,  if  p  denotes  the  pressure  at  any  intermediate  concen- 
tration over  the  acid  of  varying  composition,  is  for  dn  rnols 
transferred  : 


p 

and  the  total  work  done  is : 

Ai  =  —  2RT  |  In  *-  dtii 

•  o 

(ii.)  We  now  distil  water  from  II.  on  to  the  acid  till  we  arrive 
at  an  acid  of  the  same  composition  as  that  in  II.,  say  2(H2S04 
+  /^HoO).  The  work  done  during  the  distillation  is  : 


This  is  mixed  with  the  acid  of  II.,  and  the  2H2S04  restored. 

(iii.)  The  2H-20  may  now  be  distilled  from  II.  to  I.,  and  every- 
thing is  in  the  same  state  as  after  the  electrical  process.  The 
work  done  in  the  last  process  is  : 


The  total  work  in  (L),  (ii.),  and  (iii.)  is  : 

A  =  2RT>^  +  I  /«£»</»  - 

(     *      J0      p  o       *       J 


=  2RT  (in1*?  +  njnpt  —  njnpi  —     Inpdn] 

\    Pi  Jnl 

This  is  equal  to  the  electrical  work  2EF, 

/.  E  =  ~  (ln&  +  113/117)2  —  nilnpi  —     Inpdn}        .     (10) 
.F   \     2*1  J 

wi 


470 


THERMODYNAMICS 


E  volt. 

Density  of 

Per  cent. 

n  (mols                  p 

Solution. 

H2S04. 

H20). 

mm.  Hg. 

obs. 

calcd. 

1-496 

58-37 

69-88 

0-796 

2-29 

2-27 

1-415 

50-73 

95-16 

1-438 

2-18 

2-18 

1-279 

35-82 

175-58 

2-900 

2-05 

2-05 

1-140 

19-07 

415-8 

4-150 

1-94 

1-93 

1-028 

3-91 

2408-4 

4-575 

1-82 

1-83 

The  agreement  is  remarkably  good,  which  shows  that  the  lead 
accumulator  is  almost  theoretically  reversible,  and  the  example 
is  all  the  more  interesting  in  that  it  contains  direct  measurements 
of  the  maximum  work  (i.e.,  the  diminution  of  free  energy)  of  an 
isothermal  process  carried  out  in  two  entirely  different  ways. 

203.     Contact  Potential  Differences. 

If  an  electrolyte  AB  is  distributed  between  two  solvents,  a  and 
/3,  there  will  in  general  be  a  difference  of  potential  established 
across  the  boundary,  due  to  the  existence  of  a  double  layer 
(§  198). 

For  the  unionised  molecules: 

.         .         .         .     (1) 


where  ft  denotes  the  chemical  potential  per  mol  (§  155),  and  for 
the  dissociation  equilibria : 

(£AB  =  P>A  +  /*B)«  J  (£AB  =  £A  +  7^)0 

.'•  (/ZA  +  M,=  /*A  +  /*B)/I  •  •  •  (2) 
The  relations  (/ZA)0  =  (/*A)p ;  (/ZB)0  =  (£u)0  would  only  by  the 
merest  chance  form  the  solution  of  (2),  hence  there  will  not 
in  general  be  a  partition  equilibrium  between  the  ions  when  one 
is  established  between  the  neutral  molecules,  but  one  solvent,  say 
a,  will  contain  more  A  ions  than  corresponds  with  ionic 
partition  equilibrium.  These  will  pass  through  the  surface  of 
contact  into  /3,  and  similarly  B  ions  from  /3  to  a.  The  separa- 
tion of  the  two  kinds  of  ions  will  however  set  up  an  electrostatic 
field  across  the  boundary,  and  the  two  kinds  of  ions  collect  there 
in  two  sheets  very  close  together — in  fact,  we  have  an  electrical 


ELECTROCHEMISTRY  471 

double  layer.  Let  Ea,  E^  be  the  electrical  potentials  on  the  a 
and  /S  sides  of  the  double  layer,  then  E  =  E^  —  Ea  will  be  the 
change  of  potential  across  the  boundary.  If  the  kations  collect 
on  the  /8  side  of  the  latter,  E  is  positive. 

Now  suppose  a  small  quantity  Sc  of  electricity  passes  across 

from  a  to  /8.     This  corresponds  with  the  transport  of  ^  equiv.  of 

ions.  If  the  ions  are  univalent  this  is  also  ^  mol.  The  change 
of  thennodyuaniic  potential  is  (/iA),3  —  (/*B)«  Per  m°l 

.-.  <**>»- W-Sf    ....     (3) 

for  the  quantity  of  kation  transported. 

At  the  same  time  the  electrical  work  E&?  is  done  at  the  double 
layer,  hence,  since  in  equilibrium  : 

Sty  +  A)  =  0       .         .         .         .     (4) 


F 

or  E=w*  ^w_    .         .  .     (5) 

which  is  the  equation  of  equilibrium  of  the  kations. 

Similarly  E  =  —   ^B • p  -^ — —  .•         .         .         .  (5a) 

is  the  equation  of  equilibrium  of  the  anions,  hence 


or  (/*A  +  /ZB)«  =  (MA  +  MB)^         •          •          •     O2) 

which  we  found  above. 

In  general,  for  a  dilute  solution  : 

p.  =  T(?  +  Rl»c}    ....     (6) 
where  9  is  a  function  of  temperature  and  pressure  (§  158). 

•'•  E  =  "         (<PA)*  -  (?A).  +  B/»  (7) 


472  THERMODYNAMICS 

If  there   were   a    simple    partition   equilibrium,  without    an 
electrical  potential  difference  : 

(PA)*  =  (/*A)/S       ' '  '•     '    '-"      •         -     (8) 
or          T  [(v>}a  +  RZ»  (f A)«  1  =  T  f(9A)«  +  R/»i  (c, 


..     (9) 

Similarly  f  =  /»KB  =  Jns     .  .  (9tt) 

•"  \CB)O. 

Thence  E  =  -  *  [>KA  +  In  g£]  =  *  [foKa  +  In^] 

.    '     .         .  (10) 

The  condition  that  the  solutions  in  bulk  are  electrically  neutral, 
the  potential  difference  being  located  in  the  surface  of  separation, 
requires  that  : 

E=0.       ,.        .         .         .     (11) 

.       Ml_     (CB)]8  /.OX 

'   (cA)."~    (CB). 

If  we  add  the  two  equations  of  (10)  we  obtain  the  potential 
difference  across  the  double  layer  : 


which  is  positive  when  KB  >•  KA. 

If  we  subtract  the  same  two  equations,  we  find  : 


A  X  KB  =  2fo 

\CA.)P 

VK-xKe      ...     (14) 


The  left-hand  member  denotes  the  distribution  ratio  of  the 
kation  in  both  solvents  which  would  be  found  if  the  ions  could  be 
distributed  like  any  ordinary  solute,  and  condition  (12)  had  not 
therefore  to  be  fulfilled.  KA  and  KB  are  called  by  van  Laar,  to 
whom  the  above  formulae  are  due  (1903),  the  fictitious  distribution 


ELECTROCHEMISTRY  473 

ratios  (or  partition  coefficients)  of  the   ions ;  their  geometrical 
mean  is  equal  to  the  true  distribution  ratio. 

In  the  above  we  have  assumed  that  no  other  forces  than  the 
electrical  are  acting  at  the  surface  of  separation.  In  general, 
there  will  be  the  capillary  forces  as  well,  and  we  have  to  take 
account  of  the  influence  of  the  electrical  double  layer  in  con- 
sidering the  adsorption  of  an  electrolyte.  If  <•>  is  the  area  of  the 
surface,  <r  the  interfacial  tension,  e.  the  charge  per  unit  area,  and 
E  the  difference  of  potential,  we  shall  have  : 

S(\l/  +  A)T  =•  S(v//-  —  (rd<a  —  J&de)  =  0   .        .     (15) 
or  d(\jr  —  E?)  =  <rd(o  —  ed^Et. 

But  d(\ff  —  EC)  is  a  perfect  differential 

.     (16) 

The  charges  on  the  two  sheets  composing  the  double  layer  are 
due  to  the  accumulated  ions,  hence,  from  Faraday's  law  : 


where  y^,  y^  are  the  valencies  of  the  ions,  and  rfnA,  dnB  the 
changes  in  the  numbern  of  mols  of  the  latter  due  to  the  transport 
of  the  charge  de. 


T,   f  ±       r         B 

But  ^  =  r*'^  =  r* 

the  molar  adsorption  excesses  of  the  kation  and  anion  (§  185) 

.••rA  +  rB  =  -G,A  +  2/B)F~ 

The   total   adsorption   excess,  electrical  and  non-electrical,  is 
therefore  : 


(17) 

This  equation  is  due    to    W.    C.    McC.  Lewis,  who  wrote 


474  THERMODYNAMICS 

incorrectly,  rfU  instead  of  d^r  in  (15).  (Zeitscltr.  ^At/«/A-«/.  Chcui., 
1910.) 

204.     Single  Potential  Differences  ;  Theory  of  Nernst. 

If  a  bar  of  zinc  is  dipped  into  a  solution  of  zinc  sulphate, 
the  former  acquires  a  negative,  the  latter  a  positive  charge,  and 
a  difference  of  potential  is  established  at  the  boundary  the 
magnitude  of  which  depends  on  the  concentration  of  the  solution. 
There  is  in  fact  an  electrical  double  layer  produced,  which  is 
formed  of  negative  charges  on  the  metal  (which  has  lost  ions 
into  the  solution)  : 

Zn  +  2  ©  =  Zn 

and  a  layer  of  positively  charged  zinc  ions  with  an  equivalent 
charge.  Each  ion  leaving  the  metal  transports  a  charge  2F  per 
mol  (generally  ?/F,  where  //  is  the  valency),  and  if  we  consider 
the  transfer  of  a  quantity  de.  of  electricity  from  the  solution  to 
the  metal  we  find,  as  in  the  preceding  section,  the  equation  of 
equilibrium  of  a  metal  placed  in  a  solution  containing  its  ions  : 

£-^£de  +  me  =  ().        .        .        .     (1) 

where  /Z',  p,  are  the  chemical  potentials  of  the  ions  and  metal 
respectively,  each  referred  to  a  standard  state. 

For  a  dilute  solution  ./Z'  =  T<p'  +  RTtoc  =  prf  +RT  Inc     .     (2) 

•  .-.  E  =  -    (/*/  -  /z  +  RTtoc)    .         .         .     (3) 


or  if  we  put  BT     =  /wK        '         '         '         '     ^ 

V  =  ™lnK+W*nC  =  E°+Wl"C  '     (5) 

EO  is  the  difference  of  potential  in  a  solution  of  zinc  ions  of  unit 
concentration.  In  practice  it  is  more  convenient  to  use  the  con- 
centration f  than  c  (§  121),  and  since  in  dilute  solutions  the  two 
are  proportional,  we  can  write  : 

E  =  Eo  +  |jr  to£    •         •         .         •     (6) 

in  which  E,  E0  are  of  course  not  the  same  as  in  (5). 

EO  is  the  potential  difference  set  up  between  the  metal  and  a 


ELECTROCHEMISTRY  475 

solution  of  its  ions  containing  1  mol  per  litre  ;  it  is  usually  called 
the  electro-affinity,  and  E  is  called  the  electrode  potential.  In  a 
similar  way  we  can  find  an  expression  for  a  substance  which 
tends  to  dissolve  as  an  anion.  Thus  if  a  plate  of  platinum  is 
immersed  half  in  a  solution  of  potassium  chloride  and  half  in 
chlorine  gas,  the  trace  of  the  latter  dissolved  in  the  platinum 
dissolves  as  chlorine  ions,  leaving  the  plate  charged  positively  : 

6  =  Cf. 


The  sign  of  the  electrode  potential  is  arbitrarily  defined  as 
follows.  A  kation  electrode  (e.g.,  Zn  in  ZnS04  aq.)  is  said  to  be 
positive  when  it  is  positive  to  a  unimolar  (£  =  1)  solution  of  its 
ions  ;  an  anion  electrode  (e.g.,  C12  in  KC1)  is  said  to  be  positive 
when  it  is  positive  to  a  unimolar  solution  of  its  ions.  If  a  cell  is 
made  up  of  electrodes  reversible  with  respect  to  any  kinds  of 
ions,  its  electromotive  force  is  the  algebraic  difference  of  its 
electrode  potentials,  provided  the  electromotive  force  at  the 
contact  of  the  two  solutions,  due  to  diffusion  (cf.  Jahn,  Elektro- 
cliemie)  is  neglected. 

The  equation  shows  that  the  electrode  potential  of  a  kation 
increases  with  the  concentration,  that  of  an  anion  decreases  with 
the  concentration,  of  the  solution.  In  the  equations  (5)  and  (6) 
y  is  taken  positive  or  negative  according  as  the  charge  carried  by 
the  ion  is  positive  or  negative.  It  will  be  observed  that  the  sign 
of  E  will  agree  with  our  convention  because  we  took  the  electrical 
work  positive  for  deposition  of  the  ion,  and  this  will  tend  to 
charge  the  electrode  positively  if  it  is  a  kation  and  negatively  if 
it  is  an  anion.  The  work  done  in  depositing  1  mol  of  a  kation 
from  a  unimolar  solution  is  equal  to  its  electro-affinity,  that  for 
1  mol  of  an  anion  is  equal  to  minus  its  electro-affinity. 

Example.  —  The  electro-affinities  of  Cu  and  Zn  in  normal  (|  =  £)  solutions 
of  their  salts  are  +  0-606  and  -  0-493  volt,  referred  to  the  standard  calomel 
electrode  : 

Hg  Hg.2Cl2  »KC1  aq. 

as  zero.     The  electromotive  force  of  a  Daniell  cell  with  normal  solutions  is 
therefore  : 

0-606  -  (-  0-493)=  1-099  volt. 

Equation  (6)  can  be  written  : 
E  =  Eo  +      X  2-3026  X      log  £  =  E0  +  0-000198     log  £ 


476  THERMODYNAMICS 

In  a  normal  solution  of  a  univalent  metal,  E  =  E0 ;  in  a 
normal  solution  of  a  divalent  metal  E  =  E</  where 

Eo'  =  Eo  +  0-000198  |  log  ~ 
/.  Eo  =  Eo'  +  0-00003T. 

Thus  the  electrode  potential  of  copper  in  a  2N  solution  of  its 
ions  is  (roughly,  since  E0  refers  to  N  salt  concentration) : 

E  =  (0-61  +  0-00003T)  +  0'000198  ^  log  2 
=  0-61  +  0-000327T. 

The  electromotive  force  of  a  cell  with  solutions  of  given 
concentrations  may  be  calculated  by  subtracting  the  electrode 
potentials  so  obtained. 

Ostwald  has  assumed  that  the  equation  of  Gibbs  and  Helm- 
holtz  : 

P    ,  ,   m  <?E 

E  +  q=  +  T  -^ 

is  applicable  to  the  reactions  occurring  at  a  single  electrode,  e.g., 

Zn  +  2  0  =  Zn  +  Q 

where  Q  is  the  increase  of  intrinsic  energy  in  the  reaction. 
The  value  of  j^  has  been  measured  by  Bouty  (1879),  and 

others,  by  determining  the  electromotive  force  of  a  cell  formed  of 
two  identical  electrodes  at  different  temperatures,  e.g. : 

Zn/  ZnS04  aq.  /ZnS04  aq.  /Zn 
hot  cold 

on  the  assumption  that  the  potential  difference  between  the  hot 
and  cold  electrolytes  may  be  neglected. 
Thus,  in  N  ionic  solution  of  copper  : 

E  =  0-606 ;  T  =  291 ;  ~  =  +  0-00072 

.-.  Q  —  96540  (0-606  -  291  X  0'00072) 
=  38240  j. 

This  is  for  the  reaction  : 

iCu+  0  =  iCu. 


ELECTROCHEMISTRY  477 

The  work  done  is  EF  =  58500  j.  (for  the  Daniell  cell). 
.-.  the  latent  heat  is  58500  —  38240  =  20260  j. 

The  copper  pole  of  the  Daniell  cell  will  therefore  tend  to  cool 
itself  during  action. 

The  heat  of  ionisation  of  hydrogen  is  practically  zero,  hence 
the  heat  of  ionisation  of  a  metal  is  equal  to  its  heat  of  solution  in 
an  acid  : 

M+.H  =  M  +  |H2. 

205.     Calculation  of  Chemical    Equilibria  from  Measurements 
of  Electromotive  Forces  and  Vice  Versa. 

It  was  Helmholtz  (Ges.  Abh.,  Ill,  108;  Ostwald's  Klassiker, 
No.  124,  pp.  59jf.)  who  first  showed  how  the  free  energy  of  a 
chemical  change  could  be  calculated  from  the  electromotive  force 
of  a  voltaic  cell  in  which  the  change  occurs,  and  vice  versa. 
Thus,  in  the  gas-cell  of  Grove,  the  gases  hydrogen  and  oxygen 
are  contained,  in  contact  with  platinised  platinum  plates,  in  two 
tubes  dipping  in  dilute  sulphuric  acid.  If  the  plates  are  joined 
by  a  wire,  a  current  flows  through  the  latter  from  the  oxygen  to 
the  hydrogen,  and  the  two  gases  slowly  combine  to  form  liquid 
water.  The  large  amount  of  free  energy  which  the  gases 
possess,  which  is  dissipated  when  they  are  mixed  and  exploded, 
is  here  rendered  available  in  the  form  of  electrical  work. 

The  maximum  work  here  consists  of  two  parts : 

(i.)  The  external  work  —  RT2>,  done  by  the  atmosphere  in 
compressing  the  two  gases  as  they  are  used  up  in  the  current- 
producing  reaction ; 

(ii.)  The  work  set  free  by  the  diminution  of  available  energy  in 
the  chemical  reaction  occurring  between  the  gases  dissolved  in 
the  electrolyte  or  in  the  platinum  electrodes. 

If  we  know  the  free  energy  of  the  reaction  at  any  given 
temperature,  we  can  at  once  calculate  the  chemical  equilibrium 
at  that  temperature  from  the  equation  (§  144)  : 

AT  =  RT/nK  —  RT(2i^«Ei>,-  +  2i>,-)        .         .     (1) 

Thus,  in  the  Grove's  oxy-hydrogen  cell : 

HI,  Ha,  H3  are  the  concentrations  of  the  hydrogen,  oxygen, 
and  water  vapour  supplied  to  the  cell  and  produced  by  it, 
respectively,  in  the  reaction  : 

2Ha  +  02  =  2H20. 


478  THERMODYNAMICS 

The  water  produced  is,  however,  liquid,  hence  we  must  take  ES 
as  the  concentration  of  saturated  water  vapour  at  the  given 
temperature,  since  the  liquid  water  produced  could  be  evaporated 
to  form  vapour  at  this  concentration  without  additional  work, 
except  that  due  to  the  change  of  volume  under  the  given  pressure, 
viz.,  2RT,  which  must  now  be  omitted  fron  the  RTSr,. 

This  result  is  general  ;  the  maximum  work  for  the  production 
of  a  gas  differs  from  that  for  the  production  of  the  same  sub- 
stance in  a  saturated  solution  only  by  the  work  done  in  with- 
drawing the  gas  from  the  solution  under  the  constant  external 
pressure,  viz.,  RT  per  mol. 

The  influence  of  temperature  is  given  by  the  equation  (§  150)  : 


=  T 


AT  =  T  dT  —  RTSi^wEi  —  RTS^  +  (Const.)T  .        .  •   (2) 


or,  for  specific  heats  which  are  linear  functions  of  T  : 
AT  =  -  Qo  +  aT/wT  +  /3T2  -  RT2i/,foE, 

—  RTX^  +  (Const.)T  .         .     (8) 

If  E  is  the  electromotive  force  of  the  cell,  and  if  r  faradays  are 
transported  through  the  cell  during  the  change  for  which  the 
maximum  work  is  calculated,  we  have  : 

AT  =  rEF      ...      V.:        .         .     (4) 

since  AT  depends  only  on  the  initial  and  final  states,  and  is 
independent  of  the  particular  way  (osmotically,  electrically,  etc.) 
in  which  the  process  is  supposed  to  be  executed. 
Thus: 


rEF  =  T     ^rfT  —  RTSv^wE,  —  RTSz;,  +  (Const.)T    .     (5) 

If  we  consider  the  electromotive  force  at  a  given  temperature, 
we  have  from  (1) : 


rEF  =  RTtoK  -  RT(S^»H,  +  2^-) 

••-  E  =  ~  [inK  -  2^E,  -  2X]         .         .     (6) 

Thus  E  is,  for  fixed  concentrations  of  the  substances  used  up 
and  produced  in  the  reaction,  determined  by  the  value  of  the 
equilibrium  constant  K,  at  the  given  temperature.  The  change  of 
E  with  temperature  is  given  generally  by  the  Gibbs-Helmholtz 


ELECTEOCHEMISTRY  479 

equation  of  §  200,  or,  in  this  particular  case,  by  the  change  of  K 
as  determined  by  the  reaction  isochore  : 

tUnK        _Q^ 
dT    '  '    ET2' 

The  two  methods  are  of  course  equivalent,  both  being  founded 
on  the  same  equation,  viz. : 

AT  +  Qe  =  T  ^. 

In  equations  containing  terms  relating  to  electrical  energy, 
EF,  and  terms  relating  to  heat  absorbed,  Qc,  we  must  not  forget 
that  both  are  to  be  measured  in  the  same  units.  If  E  is  in  volts, 
and  F  in  coulombs  : 

EF  =  E  X  96540  joule, 
also  I  cal.  =  4-188  X  107  erg  =  ±'188  joule, 

•''  l  joule  =  4~W  =  °'2887  cal 
or  EF  joule  =  96540  X  0'2387  =  23046E  cal. 

Equation  (6)  therefore  gives  us  a  means  of  calculating  chemical 
equilibria  from  measurements  of  electromotive  force,  and  vice 
versa.  It  must  be  remembered  that  E  has  a  sense  only  when  it 
refers  to  a  reversible  cell ;  if  the  cell  is  not  reversible  this  simply 
means  that  no  equilibrium  can  be  set  up  at  its  electrodes  between 
the  reacting  materials. 

As  was  pointed  out  by  Ostwald  the  processes  usually  called 
oxidations  and  reductions  generally  involve  the  taking  up  of  a 
positive  charge  (or  loss  of  a  negative  charge),  and  the  loss  of  a 
positive  charge  (or  the  taking  up  of  a  negative  charge)  by  an  ion, 
respectively  : 

Fe  +  ©   =  Fe    (oxidation) 

—  G  =  Mn04 

—  ©   =  Cu      (reduction) 
+  0  =  FeCye 

If  the  ion  is  that  of  a  metal,  the  latter  must  therefore  be 
regarded  as  the  reduced  form,  or,  if  more  than  one  ion  exists,  as 
the  most  reduced  form  : 

Fe  +  3  ©  =  Fe  +  ©  =  Fe. 


480  THERMODYNAMICS 

Luther  has  put  forward  a  general  rule  relating  to  the  potentials 
and  equilibria  between  states  of  oxidation  of  a  metal.  If  the 
lowestj  intermediate,  and  highest  stages  of  oxidation  (e.g.,  Fe,  Fe; 
Fe)  are  denoted  by  I,  m,  h,  we  have  : 


the  change  of  free  energy  on  the  conversion  of  a  mol  of  the 
lowest  stage  of  oxidation  to  the  highest  being  independent  of  the 
path,  and  thus  equal  to  the  sum  of  the  two  changes  via  the 
intermediate  stage. 

If  the  electrode  potentials  between  the  metal  and  the  two  ions 
are  represented  by  E,_*  „„  E^—^  h,  and  if  a,  b  are  the  valencies, 
respectively,  of  the  two  ions  : 

m  X  F  ;  ty,_  h  =  6E,_  A  X  F 


Thus  electrode  potentials  may  be  calculated  from  oxidation 
potentials,  and  vice  versa. 

Example. — Em >/,,  measured  by  the  potential  of  the  oxidation 

electrode  Pt  |  Fe,  Fe,  is  —  0'99  volt  for  a  N  solution  of  both  ions 
(Peters,  Zeitechr.physik.  Chem.,  26, 193, 1898);  Ej_ >„,,  measured 

by  the  electrode  potential  Fe  j  Fe  is  +  0'08  volt. 

=  0-28  volt. 


.*.  if  we  mix  together  Fe,  Fe,  and  Fe.  we  shall  arrive  at  Fe,  Fe, 

+  + 

since  the  mean  state  of  oxidation  Fe  has  a  lower  potential  than 
the  highest  and  there  must  always  be  a  degradation  of  free 
energy. 

When  the  three  forms  are  in  equilibrium,  it  can  easily  be 
shown  from  the  relations 

Si/r/_>*  =  T  (J'ul  +  Elucm)  — >/,  etc. 
and  the  equation  of  equilibrium : 

vilnci  +  vzlnc?.  +  ..=  —  —  (i/j/j  -f-  i-2f2  +  .  .) 
that  E,_A  =  E,_w  =  Ew_/(. 


ELECTROCHEMISTRY  481 

In  the  case  of  gas  cells  it  is  more  convenient  to  substitute 
partial  pressures  for  concentrations  in  the  equation  : 


[/ 


Let  -x,  JTS,  .  .  .  ir,  be  the  pressures  of  the  gases  supplied  to  the 
cell,  e.g.,  H2  and  C12  at  the  electrodes  of  the  cell  : 
Ft,  H2/HC1  aq./C!2,  Pt. 

Pi,  P-2  •  •  •  Pi  the  partial  pressures  of  these  gases  in  an  equilibrium 
mixture  (e.g.,  H2  +  C12  ~  2HC1). 
From  the  relations  : 


we  find  : 


E  =  — 
>-F 


*1  "V2  •  •  ~  Sz 


where  Z»K'  =  vilnpi  + 

If  with  Nernst  and  Haber  we  omitted  Sr,  "  for  simplicity,"  we 
should  have  : 


which  is  quoted  by  Nernst  (Berl.  Bet:,  1909,  p.  247),  on  the 
authority  of  Helmholtz  (Ges.  Abhl.,  III.,  108),  but  is  not  given  by 
the  latter  in  this  form. 

Similar  considerations  apply  of  course  to  the  opposing 
electromotive  forces  of  polarisation  during  electrolysis,  when 
the  process  is  executed  reversibly,  since  an  electrolytic  cell  is, 
as  we  early  remarked,  to  be  considered  as  a  voltaic  cell  working 
in  the  reverse  direction.  In  this  way  Helmholtz  (ibid.)  was  able 
to  explain  the  fluctuations  of  potential  in  the  electrolysis  of  water 
as  due  to  the  variations  of  concentration  due  to  diffusion  of  the 
dissolved  gases.  It  must  not  be  forgotten,  however,  that  peculiar 
phenomena  —  so-called  supertension  effects  —  depending  on  the 
nature  of  the  electrodes,  make  their  appearance  here,  and  corn- 

T.  I    I 


482  THERMODYNAMICS 

plicate  the  theoretical  treatment  (cf.  Caspari,  Zeitschr.  pliyslk. 
Chem.,  30,  89,  1899;  Tafel,  Ber.,  33,  2209,  1900). 

Finally,  we  may  observe  that  measurements  of  electromotive 
force  can  often  serve  to  distinguish  which  kind  of  ions  are  really 
present  in  a  solution.  A  concentration  cell  containing  a  solution  of 
a  known  ion  with  an  electrode  reversible  to  the  latter  on  one  side, 
and  the  given  solution  with  a  similar  electrode  on  the  other  side  is 
taken.  From  its  electromotive  force,  the  concentration  of  the  par- 
ticular ion  is  calculable.  In  this  way,  for  example,  it  was  found 

+ 

that  only  a  very  small  amount  of  Ag  ion  exists  in  a  solution  con- 
taining CN  ion ;  the  greater  part  is  combined  as  a  complex  anion 
Ag(CN)2. 


CHAPTER  XVII 

THE    THEOREM    OF   NERNST 

206.     The  Statement  of  the  Problem. 

The  system  of  equations  based  solely  on  the  two  fundamental 
laws  constitutes  what  may  be  called  the  Classical  Thermo- 
dynamics. Although  perhaps  different  points  of  view  may  be 
adopted  in  the  future  in  the  interpretation  of  these  equations,  it  is 
as  unlikely  that  any  fundamental  change  will  be  made  in  this 
region  as  that  the  two  laws  themselves  will  turn  out  to  be 
incorrect. 

It  must  repeatedly  have  been  remarked,  however,  that  these 
equations  are  not  in  themselves  sufficient  to  lead  to  a  complete 
solution  of  the  problems  to  which  they  have  been  applied.  This 
arises  from  the  fact  that  they  are  differential  equations,  in  the 
solution  of  which  there  always  appear  arbitrary  constants  of 
integration  (H.  M.,  §§  73,  101,  121).  Thus,  the  relation  between 
the  pressure  of  a  saturated  vapour  and  the  temperature  is 
expressed  by  the  differential  equation  of  Clausius  (§  80)  : 


(1) 
dT        R  T2 

the  solution  of  which  is  : 

Inp  =  —  ^  y  +  const.         .         .         .     (2) 

so  that,  although  the  absolute  value  of  A,  the  latent  heat,  can  be 
evaluated  by  eliminating  the  integration  constant  at  two 
temperatures  : 

ln& 

" 

T,  ~ 

the  absolute  value  of  p  as  a  function  of  T  cannot  be  calculated. 
A  little  consideration  will  show  that  this  incompleteness  has 

i  i  2 


484  THERMODYNAMICS 

its  origin  in  the  inability  of  the  system  of  classical  thermo- 
.  dynamics  to  inform  us  as  to  the  absolute  values  of  the  intrinsic 
energy  and  entropy  of  a  system,  since  we  have  proved  that 
if  these  are  known,  and  thence  the  free  energy,  or  thermo- 
dynamic  potential,  we  are  in  possession  of  a  complete  description 
of  the  thermodynamic  properties  of  the  system. 

The  problem  which  the  classical  thermodynamics  leaves  over 
for  consideration,  the  solution  of  which  would  be  a  completion 
of  that  system,  is  therefore  the  question  as  to  the  possibility 
of  fixing  the  absolute  values  of  the  energy  and  entropy  of  a 
system  of  bodies. 

In  defining  the  intrinsic  energy  of  a  system,  Lord  Kelvin 
remarked  that  the  absolute  amount  of  energy  associated  with 
a  system  was  not  capable  of  direct  measurement,  and  might 
be  very  large  —  in  fact  we  could  not  be  sure  that  it  was  not 
infinite.  The  newly  developed  theory  of  Relativity  shows  that 
the  energy  associated  with  a  body  at  rest,  if  we  leave  out  of 
account  external  actions  such  as  pressure,  is  equal  to  the  mass  of 
the  body  multiplied  by  the  square  of  the  velocity  of  light  in 
vacuum  —  an  enormous  magnitude.  Since,  as  we  shall  see  later, 
the  ambiguity  does  not  really  involve  the  absolute  value  of  the 
energy,  we  shall  not  encumber  our  equations  with  this  colossal 
additive  constant,  and  we  shall  continue  to  measure  the  energy 
from  an  arbitrarily  selected  standard  state. 

With  the  entropy,  however,  the  case  is  quite  otherwise,  and  we 
shall  now  go  on  to  show  that  as  soon  as  we  are  in  possession  of  a 
method  of  determining  the  absolute  value  of  the  entropy  of  a 
system,  all  the  lacunae  of  the  classical  thermodynamics  can  be 
completed.  The  required  information  is  furnished  by  a  hypo- 
thesis put  forward  in  1906  by  W.  Nernst,  and  usually  called 
by  German  writers  "  das  Nemstsche  W&rmeiheorem"  We  can 
refer  to  it  without  ambiguity  as  Nernst'  s  Theorem.^ 

207.     The  Theorem  of  Nernst. 

The  entropy  of  a  condensed  chemically  homogeneous  substance 
vanishes  at  the  zero  of  absolute  temperature: 


By  "  condensed  "  is  meant  solid  or  liquid  ;  by  "  chemically 
homogeneous  "   is   meant   a  pure  substance,    element  or  com- 


THE    THEOREM  OF  NEENST  485 

pound,  as   distinguished   from   a   mixture   or  solution.     Terms 
relating  to  such  substances  will  be  enclosed  in  square  brackets. 

208.     Specific  Heats  and  Coefficients  of  Expansion. 

If  the   independent   variables  are  p  and    T,   we   have   quite 
generally  for  a  homogeneous  substance  : 


=  f (1) 

M  _          fr\  (2) 

(3) 


-;* 


The  integral  does  not  furnish  the  absolute  value  of  »,  the 
entropy,  because  the  lower  limit  is  undetermined.  If  this  is 
regarded  as  fixed,  the  integral  with  various  upper  limits  gives 
the  values  of  the  entropies  referred  to  this  arbitrary  standard 
state,  and  the  differences  between  these  values  and  any  one  of 
them  referred  to  this  arbitrary  standard  state  will  be  the  values 
of  the  entropies  referred  to  the  new  standard  state  (cf.  §  42). 

If  we  introduce  Nernst's  theorem  : 


we  can  however  transform  the  indefinite  integral  into  a  definite 
integral  with  the  lower  limit  zero  (cf.  §  192),  since  s  is  positive 
and  vanishes  for  T  =  0  : 


•JP 


T  -  (5) 

'0 

Since  for  T  =  0  the  integral  does  not  become  infinite,  it  follows 
that: 

[<gT=0  =  0 (6) 

This  remarkable  result  has  been  verified  by  experimental 
measurements  of  specific  heats  at  very  low  temparatures,  viz.,  in 
liquid  air  and  liquid  hydrogen  (cf.  references  in  Chap.  I.).  It 
was  formerly  believed  that  the  specific  heats  of  solids  approached 
small  positive  limiting  values  at  the  absolute  zero,  but  the  form 
of  the  curve  at  very  low  temperatures  alters  appreciably,  and  it 
may  be  inferred  that  the  specific  heat  is  vanishingly  small  at 


486  THERMODYNAMICS 

very  low  temperatures.      In  the  case  of  the  diamond,  no  heat 
capacity  at  all  can  be  detected  in  liquid  hydrogen. 

0.  m  /3/A      idi'\2     /e  ^ 

Since  c,,  —  ce  =  —  1 I  gM    •  1^-1       (§  64) 

it  follows  that,  at  the  absolute  zero : 

since  (om )  is  shown  below  to  be  zero  at  the  absolute  zero,  and 

Up-J  was  found  by  Griineisen  <]>  to  be  nearly  constant. 

cp,  cr  are,  of  course,  functions  of  temperature,  and  we  shall 
see  later  how  the  form  of  these  functions  has  been  determined, 
partly  by  means  of  atomistic  considerations,  partly  empirically. 

Equation   (5)   holds   for  all    values   of  p,   and   hence   if  we 
differentiate  with  respect  to  p  and  compare  with  (2)  we  have  : 
T 


P)      7T  fr>\ 

4/TfZT=~V8TJ, 


But 


°r      'do^lf' 


the  coefficient  of  expansion  of  the  substance  also  vanishes  at 
absolute  zero.  The  experimental  evidence  for  this  is  not  so 
satisfactory  as  that  relating  to  specific  heats,  but  it  is  at  least 
certain  that  the  coefficients  of  expansion  of  metals  diminish 
rapidly  at  low  temperatures.  The  empirical  relation  : 

specific  heat 


coefficient  of  linear  expansion 


=  constant    .         .         .         .     (9) 


THE    THEOREM  OF   NERNST  487 

verified   for   some   metals   by  Griineisen  also   strengthens  the 
conclusion. 

209.     Transition  Points. 

In  Chap.  VII.  we  have  considered: 

(i.)  changes  of  physical  state, 
(ii.)  allotropic  or  polymorphic  changes, 
(iii.)  chemical  reactions 

which  occur  at  a  transition  point. 

A  particularly  good  example  of  the  second  type  is  the  tran- 
sition from  rhombic  to  monoclinic  sulphur.  2 

The  equation  of  equilibrium  is  (§  97)  : 


/.     MI  —    si      7>ri  =  u.2  — 

or  «'i  —  T.S!  =  »'2  —  T-s'o 
where  n-  =  u  +  7>r. 

Thence  T  (s2  —  *0  —  (n-a  —  «'i)  =  0. 

/•T  /.T 

But 


/•T 

sl=\  ^ 

•' 


/.T 

...     T   I  C>>-  ~  tVl  r/T  —  A  =  0       .        .        .     (10) 

•  0 

where  A  =  /r-2  —  ifi  =  latent  heat  of  transition. 

The   equation,   with   known   <-/v  <>,,,>    an(l   A»   determines   the 
transition  temperature  for  (ft). 


-'•    ^-^  =  (cf-  §§93,97) 


T-     =  0        .        .        .    (11) 
an  equation  which  will  be  generalised  in  the  sequel. 


488  THERMODYNAMICS 

According  to  Bronsted,  for  the  sulphur  transition  : 
A  =  1-57  +  115  X  1(T5T2 

|j  =  2-3  X  1CT5T 

/.     T  =  369-5  (obs.  =  869-4). 

We  observe  that  in  the  empirical  power  series  for  A  given 
by  Bronsted  there  is  no  term  with  T.  It  can  be  shown  that 
this  is  peculiar  to  all  reactions  covered  by  the  theorem  of 
Nernst.  For  let  us  assume  that  the  heat  of  reaction  is  expres- 
sible by  such  a  series  : 

Q  =  A  +  BT  +  CT2  +  DT3  + (12) 

where  Q  refers  either  to  constant  volume  or  constant  pressure : 

Q,  =  U2  -  Ui 
or  =  Wa-Wi. 


Then  ^     =  r;  -  T,  =  0        .        .         .    (13) 

L          J    T  =  0 

and  ~jf     =  iy  —  r,  =  0         .         .         .     (14) 

since  r,  =  2vGv  =  0 

TyJ  =  ^vCtl  —  0. 

Hence  there  can  be  no  constant  term  in  the  derived  series, 
i.e.,  the  coefficient  of  T  in  the  series  (12)  must  be  zero. 

The  expansion  of  Q  in  a  power  series  must  therefore  have  the 
form  : 

Q  =  Qo  +/3T2  +  7T»  +  .  .  . 

where  6  =  Sr/3.- 


and  the  expressions  for  the  molecular  heats  of  the  different  con- 
densed phases  are  of  the  form  : 

Cc.  =  a,  +  2&T  +  3%T2  +  .  .  .  etc. 
i.e.,  Cr.  =  2&T  +  37/r2  +  .  .  . 

since  (L.  =  0  for  T=0  /.  a(  =  0. 


THE  THEOREM:  or  XERNST  439 

If  the  reaction  occurs  at  constant  volume  : 


and  Q,.  =  U2  -  LT!  =  *2  +  TS2  -  (*!  +  TSi) 

/.  ^  =  5-^t=^  +  (8,  -  SO,  and  hence, 


?T 

since  [S2]  T  =  o  =  [Si]  T  =  o 

If  we  put  —  Q,.  =  Qr 

=  diminution  of  intrinsic  energy 
=  heat  of  reaction  (erohed) 

we  can  write  (15)  in  the  form  : 
~dQ0 


=  0         .        .     (15a) 

T  =  0 

which  is  the  form  adopted  by  Nernst.  3  This  mode  of  expression 
leaves  open  the  question  as  to  the  value  of  the  specific  heats 
at  T  =  0,  and  simply  requires  that,  at  this  temperature,  the 
molecular  heat  of  a  compound  shall  be  additively  composed 
of  the  atomic  heats  of  its  elements,  i.e.,  the  Xeumann-Kopp 
rule  is  strictly  followed  at  the  absolute  zero.  This  must,  how- 
ever, be  the  case,  since  all  the  atomic  and  molecular  heats  are 
zero. 

If  the  pressure  is  constant  we  have  similarly  : 


where  AT'  is  the  maximum  work  under  constant  pressure,  and 

Q,=  ~  Qr 

Then,  from  the  equations  : 


A  f       n   - 
AT   ' 


490 

we  find  : 


THERMODYNAMICS 


Thus  if  we  take  T  as  abscissa,  and  Q,  AT,  as  ordinates, 
the  two  curves  meet  tangentially 
and  parallel  to  the  temperature 
axis  at  T  =  0  (Fig.  88).  This 
simply  shows  that  the  quantities 
AT  and  Q  approach  more  and  more 
closely  as  T  approaches  zero ; 
near  the  absolute  zero  they  coin- 
cide, and  the  value  then  remains 
constant  down  to  the  absolute 
zero.  Other  views  were  previously 
.  held.  <4> 

If  we  suppose  the  specific  heats 
are   functions   of   T   of  the    form 


C  =  2/3T  +  37T2  +' 


then  : 
also 


[Q]  =  [<2o]  +  /3T2  +  7T3  + 


(19) 


[A,]  =  [Q]  +  T 


.        [(21  _ 

-       -- 


8[AT] 
r?f 


dT  T2  r 

.-.  [AT  =  [Q0]  -  /3T2  -  .'.  7T8  H 

Nernst  uses  the  two  very  simple  equations  : 


(20) 


(19a) 
(20a) 


for  reactions  occurring  in  condensed  systems  between  chemically 
homogeneous  substances.  The  possibility  of  calculating  [AT], 
and  thence  transition  points,  etc.  ([AT]  =  0),  from  purely 
thermal  magnitudes  is  then  especially  clear.  This  method  may 
also  be  extended  to  reactions  in  which  gases  occur,  e.g.,  in  the 
dehydration  of  salts  <5> : 

CuS04  .  H20  -*  CuS04  +  H20  (vap.) 
provided  we  imagine  the  reaction  to  occur  between  the  substance 


THE  THEOREM  OF  XERNST        491 

and  ice  ;  the  heat  of  fusion  is  then  to  be  subtracted  from  [O], 
and  [AT]  is  the  work  done  in  transferring  n  mols  of  H20  in 
the  form  of  vapour  from  ice,  at  the  vapour  pressure  ir0,  to  the 
hydrate,  at  the  dissociation  pressure  IT,  viz.  : 


-0  can  be  calculated  from  Scheel's  formula  of  §  95. 

Example  1.  —  If  the  specific  heats  of  the  solid  and  liquid 
forms  are  linear  functions  of  temperature,  show  that  the 
melting-point  is  determined  by  dividing  the  latent  heat  of 
fusion  by  the  difference  between  the  specific  heats  of  the  solid 
and  liquid  forms  at  the  melting-point  (cf.  Tammanu,  Kri/st. 
und  Schmelz.,  p.  4*2). 

Example  2.  —  Discuss  the  possible  forms  of  the  >\T]  and  \Q~: 

> 
curves  according  as  /3  =  0  (cf.  T.  \V.  Richards,  Proc.  Amer.  Acad., 

< 

1902;  Zeitschr.  i)hysik.  Chem.,  40,  169,  597,  1902  ;  42,  129,  1903). 
These  relations  are,  however,  certain  only  at  low  temperatures 
(cf.,  Nernst,  Ber.  Berl.  Akad.,  1909,  247). 

210.     Gases  ;  Evaporation  and  Sublimation 

When  we  come  to  deal  with  gases  or  vapours,  we  pass  at  once 
out  of  the  region  of  direct  applicability  of  Xernst's  theorem.  If 
we  assume,  approximately,  that  the  specific  heat  of  the  gas  is 
constant  over  a  small  range  near  the  absolute  zero,  we  have 
(§  79)  : 

.s  =:  s0  +  fr/;iT  -f-  rlnr 

and  this  shows  that  the  entropy  of  a  gas  by  no  means  vanishes 
for  T  =  0,  like  that  of  a  liquid  or  solid,  but  on  the  contrary, 
would  become  negatively  infinite  : 

ST=O=  -  oo         .          .          .          .     (1) 

Since  the  specific  heat  of  a  gas  can  always  be  expressed  in  the 
form  : 

cc  =  a  +  6T  -f  cT2  +  •  • 

where  a  is  finite,  or  cc  =  a  +  F  (t),  where  F  (o)  =  o,  but  a  is 
finite  if  the  law  of  equipartition  of  energy  holds  good  for  the 
molecules  of  the  gas  at  T  =  o  (cf.  §  213),  this  result  is  general. 
If  a  gas  is  in  equilibrium  with  a  solid  or  liquid  of  the  same 


492  THERMODYNAMICS 

composition,  and  both  are  chemically  homogeneous,  the  chemical 
potentials  of  the  substance  in  both  phases  are  equal  (§§  106, 157) : 

/*       =     M (2) 

vapour  condensed 

At  very  low  temperatures  the  pressures  of  the  saturated 
vapours  of  liquids  and  solids  are  very  small,  and  since  the  devia- 
tion of  an  actual  gas  from  the  laws  of  ideal  gases  becomes  all  the 
less  the  smaller  is  its  density  (§  70),  we  can  safely  assume  that 
the  vapour  obeys  the  gas  laws. 

If  [m],  m  are  the  molecular  weights  of  the  condensed  form  and 
of  the  vapour,  respectively,  and,  [<£],  </>  their  thermodynamic 
potentials  per  mol : 


Also  (§  79) : 

d>  =  U0  -  TS0  +   I  C»dT  -  T    I  ^  dT  -  ET  (Inv  -  1) 


f  (VT  - 


=  U0  -  TS0  +    I  (C,  +  B)<ZT  -  T        Jv  j"  ™>  dT 

+  RTlnp  -  ETln  5 

C  C^ 

=  Uo  -  TS0  - 


-  T  j  ^  dT 


-  ET/w  ~    .        .         .     (4) 


and    [<^]  =  ([U]  +  p{\])  -  T[S]  =  [W]  -  T   \-dT      .     (5) 

Thence  : 
U0  -  TS0  +  Jc/T  -  rij^  dT  +  ET//^  -  BT/w  5 


THE    THEOREM  OF   NERNST 

U0  +  f  C.//T  =  W 


493 


But 

the   latent  heat  of  volatilisation   referred   to   1  ruol  of  vapour, 
hence : 


—  TS0  —  T     y  dT  +  RTlnp  —  RT?»  ^ 

J 

For     T  =  0  :         A  =  A0  =  U0  -  a  [W0] 


R=_aT  f[C 
T 

•'O 


CUT  -  a      [CJrfT 


/.  Ao  -  TSo  +    I    C,//T  -  T  I  ^  rfT  +  RT/i^>  -  RT/w   ^ 


"T  XT 

[CU/T  -  T  ' 


11  w 

or,  since  (§  58)  :          A    =  A0  +  {(C,,  —  a  [Cp]  )rfT 

-^JT-al  ffiicTcU-.    .     (7) 


where 

J-i 

At  low  temperatures  the  value  of 


»  =  |p  +  fa  g  =  const.  .         .     (8) 


rT  rT 

!    a       rr  i  //T  a     ^  ,/T 

IT    [C"]'/T  ~R!  -Y~dT\ 

J0  Jo 


is  negligibly  small,  since  [C^  =  0  =  0,  hence  at  low  temperatures  : 


top  =  —  wr  ~ 


•     (9) 


494  THERMODYNAMICS 

All  magnitudes  depending  on  the  properties  of  the  solid  have 
now  vanished,  and  <o  will  therefore  not  depend  on  the  condition 
of  the  latter. 

If  we  assume  the  specific  heat  of  the  vapour  to  be  independent 
of  temperature  : 

Cp  =  const., 


- 


where  P  =  w  —  =*•'  =  const.         .         .         .     (11) 

XI 

If   the   specific   heat   of   the   vapour   is   a  linear  function  of 
temperature  : 

CP  =  C0  +  R  =  a  +  R  +  2/3T, 


(12) 


where  /  =  »  —  —  (13) 

it 

The  magnitude  w,  depending  only  on  the  specific  chemical 
composition  of  the  vapour,  may  be  referred  to  as  the  primitive 
chemical  constant  ;  p  and  i  are  derived  chemical  constants. 

Equations  (10)  and  (12)  have  been  discussed  by  Nernst  (Vcrh. 
Deutsch.  Phys.  Ges.,  11,  818,  1909  ;  12,  565,  1910  ;  cf.  Recent 
Applications  of  Thermodynamics  ;  Planck,  Thermodynamik,  8 
Aufl.,  275). 

The  chemical  constants  may  therefore  be  determined  directly 
by  the  measurement  of  vapour  pressures,  especially  at  low 
temperatures.  Equation  (9),  which  is  more  general,  shows  that 
the  chemical  constant  is  determined  for  a  homogeneous  gas  as 
soon  as  we  know  A,  and  Cp,  as  functions  of  temperature,  and  the 
vapour  pressure  at  one  temperature.  These  data,  especially 
vapour  pressures  at  very  low  temperatures,  are  not  very  well 
known  at  present,  and  some  other  method  must  therefore  be 
used  in  the  determination  of  the  chemical  constant.  Several 
such  methods  have  been  proposed  by  Nernst  (he  .  cit.  ;  cf  .  also 
Haber,  Thermodynamics  of  Technical  Gas  Reactions,  pp.  88  —  96  ; 
Weinstein,  Thermodynamik  und  Kinetik  III.,  2,  pp.  1064—1074). 


THE  THEOREM  OF  NERNST        495 

211.     Determination  of  the  Chemical  Constant. 

(i.)  By  direct  comparison  of  the  vapour-pressure  curve  with 
the  equation : 

lnp=-±  +  ^^lnT  +  |  T  +  i    .         .     (1) 

According  to  Nernst,  the  vapour-pressure  curves  of  different 
substances  can  be  represented  over  a  considerable  range  by 
means  of  the  empirical  equation  : 


=  1T5  log 


(2) 


where  a\  is  a  constant,  although  he  does  not  give  any  numerical 
data  in  support  of  this  assertion.  0.  Brill  ""  has  shown  that  this 
equation  does  not  well  represent  the  vapour-pressure  curve  of 
ammonia  at  low  temperatures. 


Thus:      logp=-        5  +1-75 

+  (l-424a'  +  \ogpK  -  1-75  log  TK)    .  (2«) 

If  we  multiply  (1)  throughout  by  0'4343  to  convert  to  common 
logarithms,  and  compare  terms  with  (2)  we  find  : 

0-4343A0  =  a'RTK  =  2a'TK 
0-4343  £  =  1-75  /.  -4343a  =  T75  x  2  =  3'5  =  const.     (8) 

0-4348/9  =  - 


and  0'4343i  =  l'424a'  +  logpK  -  1*75  log  TK  .         .  (3a) 

In  this  way  a'  and  i  can  be  calculated  for  various  gases  and 
vapours. 

(ii.)  The  second  method  is  an  application  of  the  Clapeyron- 
Clausius  equation  : 

A  =  T  (V  -  v')  ^  •          •          •          .     (4) 
According  to  S.  Young  the  equation  : 


496  THERMODYNAMICS 

holds  with  close  approximation  ;  for  states  widely  removed  from 
the  critical  this  goes  over  into  the  simple  equation  : 

v  =  — 
~   p 

usually  assumed  in  the  integration  of  (4). 

Nernst  now  represents  X  as  a  function  of  temperature  by  the 
equation  : 


A  =  (A0  +  aT  +  fcT2)       -          .         .         .     (6) 
/.  by  integration  of  (4)  : 

(Ao  +  aT  +  I'P)  =  RT2  ^ 
Inp  =        -  $•  +  a  InT  +  7/l    +  A      .         .     (7) 


where  A  is  an  integration  constant. 
By  comparison  of  this  with  (1)  : 

a  =  a,  b  =  /3,A  =  i  .    .      .  $    .         .     (8) 

The  magnitudes  A0  and  b  can  be  determined  from  (6)  if  we 
know  corresponding  values  ot  p,  T,  and  A. 

Later  (cf.  Berl.  Ber.,  1909,  247)  he  proposed  the  equation  : 
A  =  AO  +  aT  —  eT2 


where  Cp  =  molecular  heat  of  vapour;  [C]  =  molecular  heat 
of  condensed  form.  According  to  the  kinetic  theory  of  gases, 

Cp  <£  |  R  <fc  4-962   for   a   monatomic  gas,  whilst  [C]   sinks  to 

very  small  values  at  low  temperatures.  Nernst  takes  a  =  3  -5 
for  all  substances,  as  already  mentioned.  It  is  evident  from 
the  new  assumption  that  [C]  =  0  when  T  =  0,  that  a 
should  be  4*962,  which  is  the  value  proposed  by  Thiesen.  The 
introduction  into  the  Clapeyron-Clausius  equation  then  leads  to  : 


where  C  =  0'4343i     ....  (10) 

C  is  Nernst's  chemical  constant. 

This  part  of  the  theory  evidently  stands  in  need  of  revision. 


THE    THEOREM  OF   NERNST  497 

The  values  of  i  calculated  from  (3)  and  (8)  do  not  agree  very 
closely,  and  it  would  appear,  as  Weinstein  (loc.  cit.  1068)  remarks, 
that:  "Although  the  calculations  undoubtedly  establish  the 
legitimacy  of  the  system  of  equations,  the  great  uncertainty  in 
the  numerical  determination  of  the  decisive  magnitudes  forms  a 
practical  defect  which  will  only  be  removed  by  observations  over 
very  wide  intervals  of  the  variables."  Any  discrepancy  between 
the  results  of  actual  observations  of  equilibria,  and  those  calcu- 
lated by  means  of  Nernst's  "  chemical  constants,"  need  not,  in 
the  present  state  of  uncertainty  of  the  latter,  cause  any  great 
alarm.  Nernst  himself  apparently  regards  the  constant  <o, 
obtained  from  vapour-pressure  measurements,  as  the  most 
certain,  and  the  others  as  more  or  less  tentative. 

The  vapour-pressure  equation  (22)  has  been  verified  for  brorn- 
and  iodo-naphthalene  by  L.  Rollo  (7'  (1909),  and  for  toluene, 
naphthalene,  and  benzene  by  J.  T.  Barker  (1910).  The  latter 
finds  that  the  solid  and  liquid  states  give  the  same  chemical  con- 
stant, which  is  in  agreement  with  Xernst's  theory. 

(iii.)  The  method  for  the  evaluation  of  the  chemical  constants 
which  has  been  most  used  is  the  determination  of  gaseous  equilibria, 
which  is  considered  below. 

Planck  (loc.  cit.  276)  has  observed  that  the  point  on  which  the 
whole  matter  turns  is  the  establishment  of  a  characteristic 
equation  for  each  substance,  which  shall  agree  with  Nernst's 
theorem.  For  if  this  is  known  we  can  calculate  the  pressure  of 
the  saturated  vapour  by  means  of  Maxwell's  theorem  (§  90). 
He  further  remarks  that,  although  a  very  large  number  of 
characteristic  equations  (van  der  Waals',  Clausius's,  etc.)  are  in 
existence,  none  of  them  leads  to  an  expression  for  the  pressure 
of  the  saturated  vapour  which  passes  over  into  (9)  §  210,  at  very 
low  temperatures.  Another  condition  which  must  be  satisfied  is 


and  the  establishment  of  a  suitable  characteristic  equation  — 
granted  that  this  is  possible  (§  114)  —  is  one  of  the  most  important 
problems  of  the  future. 

212.     Equilibrium  in  Gaseous  Systems. 

Although  the  theorem  of  Xernst  does  not  directly  apply  to 
gaseous  equilibria,  it  is  possible  by  a  very  simple  expedient  to 

T.  K   K 


498  THERMODYNAMICS 

introduce  its  consequences  into  an  equation  for  homogeneous 
gaseous  equilibrium,  and  also  to  determine  the  maximum  work 
of  an  isothermal  gas  reaction. 

The  equations  given  by  Nernst  depend  on  the  assumption  of 
the  form  : 

CP  =  a  +  2/3T  +  37T2  +  .  . 

for  the  specific  heats  ;  Planck  (Thermodynamik,  3  Aufl.,  p.  277) 
assumes  the  specific  heats  of  the  gases  to  be  constant.  We  shall 
deduce  the  equations  quite  generally. 

For  this  purpose  we  take  temperature  and  pressure  as 
independent  variables,  and  write  down  the  equation  for 
equilibrium  (§  143)  : 

2(<p,  +  R/MC,>,  =  o  .'C;;,:      .  *    .    (l) 

where  the  summation  extends  over  all  the  components  and  9^ 
is  a  function  of  temperature  : 


fcv 

-  rp 


w,  is  the  chemical  constant  (§  210,  eqn.  (9)  ). 
Thus  : 


(2) 


(8) 


;"+    C»dT  =  W 


But  U;"+    C»dT  =  W,  .        .        .        .     (4) 


and  2r,W,  =  Q,      ;        .        .        .     (5) 

the  heat  of  reaction  at  constant  pressure,  hence  : 


v  7  Q*  ,  2^fc;v/T     1 

2>./»c,.  =  -  ^L  _|_  _« I  _^_ .  v, 


(6.) 


THE    THEOREM   OF    NERNST  499 

If  we  put 

Qp  =  Qo  +  2*(  I  CjjMT  .        .        .        .    (7) 

Jo 
where  QJt  =  heat  of  reaction  at  absolute  zero  : 

-     ~      CWT  </T  -  ^7"' 


Also  2,i>ilnCi  =  InK,  where  K  ==/  (j>,T)  is  the  equilibrium 
constant. 

Equation  (8)  is  the  most  general  form  of  Gibbs's  equation 
(J  148),  and  goes  over  into  the  latter  when  Cjj'  =  const. 

If  we  compare  (8)  with  equation  (2)  of  §  148,  putting 
const.  =  Q  in  the  latter,  we  see  that  the  indeterminate  constant 
of  the  equilibrium  equation  is  now  completely  determined  as  the 
sum  of  the  chemical  constants  of  the  interacting  gases  : 


and  may  be  found  from  investigations  of  the  vapour  pressures  at 
low  temperatures.  The  chemical  equilibrium  at  any  temperature 
may  then  be  calculated  without  a  single  measurement  of  its 
value. 

If  we  use  the  equation 

C,  =  Cr  +  R  =  a  +  R  +  2/3T 

for  the  specific  heat  of  each  gas,  the  equation  (8)  goes  over  into  : 
_^+M^±B),,,T  +  ^T 

2,,, 


where       a  = 


.     (9) 


Full  details  of  the  application  of  equation  (9)  are  to  be  found 
in  the  monographs  referred  to  below  ;  it  will  be  sufficient  here  to 
work  out  one  case  in  detail,  and  we  shall  take  the  dissociation  of 
carbon  dioxide  : 


K  K  2 


500  THERMODYNAMICS 

For  convenience  we  replace  concentrations  by  partial  pressures  : 


T  +  2™   -     (10) 


P2CO'2 

Let  g  =  percentage  of  dissociation,  i.e.,  the  number  of  mole- 
cules per  100  which  are  broken  up  at  a  given  temperature,  then 
/;/100  =  7,  the  fraction  of  dissociation. 

27  7  _  2  (100  —  7) 

pcc  ~  200  +  7 '  P°2  ~200  +  7 '  ^co'2  ~~     200  X  7 

•••K'=r-.rff-^y 

v2  +  ioo/  V1     ioo/ 

=  ^  when  7  is  small 
.'.  log  K'  =  3  log  7  —  log  2  =  3  log  7  —  0'3. 

We  multiply  each  term  by  0'4343  to  transform  natural  to 
common  logarithms,  and  obtain  finally  : 

n.  Q 

log  K'  =  — 


where  C,  =  0*4843^ 

According  to  the  assumptions  of  §  211 : 

0-4848  *y<(%+B>=  1-75* 
±\ 

and  yS  is  calculated  by  means  of  another  arbitrary  assumption, 
viz.,  that  the  molecular  heat  of  a  gas  is  of  the  form  : 

CJ?  =  8-5  +  2/3/1 


THE    THEOREM  OF   NERNST 


501 


- 


Again 

QP  =  Qo  +  8-5*T  +  /3T2      .         .         .     (12) 

from  which  Q0  is  calculated  from  the  observed  heat  of  reaction  at 
any  one  temperature. 

The  data  are  (cf.  Nernst,  Recent  Applications)  : 

C02C,       =10-05 
€0,020,=    6-9 
Q,  =  136000  cal.  at  T  =  290 
.-.    Qo  =  135000,  and 

Qp  =  135000  +  3-5T  -  0-0030T2. 

The  chemical  constants,  C,-,  are  : 

CO  =  3-6  ;  02  =  2-8  ;  C02  =  3-2 
/.  ^d  =  2X3'(5  +  2-8  —  2  X  3-2  =  3'6 

(The  value  for  CO  given  by  Nernst  is  3'6  ;  according  to  Weigert, 
Abegg's  Handbuch  der  anorg.  Chem.,  Art.  Kohlenstott',  the  value 
2'6  is  more  probable.) 

Thus,  finally  : 


29600  .    t  __ 
3  log  7  = —  +  1-75 


T  —  0-00066  T  +  3-9 


y  =  lOOy. 

T  (obs.  Nernst  and 
Wartenberg). 

T  (calc.). 

0-00419 
0-029 

1300 
1478 

1369 
1552 

In  some  cases  the  variation  of  specific  heat  with  temperature 
and  the  difference  of  specific  heats,  are  small,  and  the  terms  with 
T  and  log  T,  respectively,  may  be  omitted  : 


log  K'  =  — 
log  K'  =  - 


4-571T   '  4-571 
since  Qo  in  these  cases  is  approximately  equal  to 


502 


THEBMODYNAMICS 


The  table  of  chemical  constants  given  by  Nernst  (Berl.  Her. 
1909,  p.  247)  is  : 


H2 

l-6(2) 

NH3 

3-31 

S2U)            3-15 

CH4 

2-5 

H20 

3-7 

N2 

2-6 

CC14 

3-1 

02 

2-8 

CHC13 

3-2 

CO 

3-6 

HC1 

3-0 

C12 

3-2 

H2S 

3-0 

C02 

3-17 

I2 

4-2 

(1)  Preuner  and  Schupp,  Zeitschr.  physik.  Chem.,  68,  2,  129, 
1909. 

(2)  Nernst,  Zeitschr.  Elektrochem.,  15,  18,  687,  1909. 

213.     Solid  and  Liquid  Solutions. 

The  theorem  of  Nernst  applies  only  to  chemically  homogeneous 
condensed  phases ;  the  entropy  of  a  condensed  solution  phase 
has  at  absolute  zero  a  finite  value,  owing  to  the  mutual  presence 
of  the  different  components. 

It  may  reasonably  be  assumed  that  the  terms  in  the  expression 
for  the  entropy  which  depend  on  the  temperature  diminish,  like 
the  entropy  of  a  chemically  homogeneous  condensed  phase,  to 
zero  when  T  approaches  zero,  and  the  entropy  of  a  condensed 
solution  phase  at  absolute  zero  is  equal  to  that  part  of  the  expres- 
sion for  the  entropy  which  is  independent  of  temperature,  and 
depends  on  the  composition  (Planck,  Thermodynamik,  3  Ann1., 
279). 

In  the  case  of  a  dilute  solution  this  is  (§  185)  : 

S  =  -  RSvJm;     ....     (1) 

In  the  case  of  a  solution  of  moderate  concentration  we  may 
perhaps  assume  the  same  expression  (cf.  van  Laar,  Thermo- 
dynamik und  Chemie ;  Thermodyn.  Potential ;  Planck,  loc.  cit.), 
whenever  the  solutions  can  legitimately  be  considered  as  brought, 
by  suitable  changes  of  temperature  and  pressure  with  unchanged 
composition,  into  ideal  gas  mixtures  (§  185). 


THE  THEOREM  OF  NERXST        503 

The  entropy  of  a  condensed  homogeneous  solution  (e.g.,  an 
isoniorphous  solid  solution)  is  then  : 


.    (-2) 


If 


this  expression  goes  over  into  that  for  a  chemically  homogeneous 
substance. 

Corolla  rii  1.  —  The  specific  heat  of  a  condensed  solution  vanishes 
at  absolute  zero. 

Corollary  2.  —  The  coefficient  of  expansion  of  a  condensed  solu- 
tion vanishes  at  absolute  zero. 

214.     Heterogeneous  Equilibria. 

For  a  system  composed  of  any  number  of  condensed  chernically 
homogeneous  phases  and  a  homogeneous  gas  phase  we  have  the 
symbol  : 

H«"/«o"   |    HO'  "/Mo"'    !•••• 


where  all  letters  with  zero  suffix  refer  to  condensed  bodies. 
In  the  isothermal-isopiestic  change  : 

ono   '•  &HO"  '•  ono'"  :  .  .   .  :  &HI  :  S«2  :  on3  :  •  •   • 

=  VQ   :  v0"  :  VQ"  :  .   .  .  :  n  :  r-2  :  ^3  :  •  •  • 

we  have,  as  the  condition  for  equilibrium  : 

gef>  =  0     (fy  =  0,  8T  =  0)      .         .         .     (1) 
But        4>  =  HO'<£O'  +  "o"^>o"  -f  .  .  .  +  HI/LI  +  11-2^-2  +    •         •     (2) 

where  ^>o'  =  thermodynamic  potential  of  1  mol  condensed  homo- 
geneous t'-th  phase 

=  W  -  T«0;  +  7«V  =  «V  -  W  ....  (3) 
and  p>i  =•  chemical  potential  of  t'-th  component  of  gas  phase 

=  T(9,-  +  Elnc,)    .......     (4) 

For  a  very  small  isothermal-isopeistic  change,  with  changes  of 
composition  of  the  separate  phases  vanishing  in  the  limit  (§  142): 

.  .  =  0 

=0       -     (5) 


504  THERMODYNAMICS 

where  the  first  summation  extends  over  the  ;•  condensed  phases, 
and  the  second  over  the  i  components  in  the  gas  phase. 


=     ^p 


But  s0(r),  i.e.  [s0(r)]  =  dT    .         ,        .     (6) 

by  Nernst's  theorem,  and 

9<  =  ^  -  Sff>  +     *  \  C)|WT  -  |  °|  dT  +  E  (inp  -  to  *, 

Tin/i        rnu) 

=    T    ~J  T  </T  +  ™np 
from  §  212,  hence  : 

,)  -  TS 


+  RTv/wj)  —  RTS^w,-  =  0    .         .     (8) 
But  SC^'iroW  +  »,W,)  =  Qp 

the  heat  of  reaction  at  constant  pressure, 
and 


—  2iv»<   .     (9) 

If  we  put  Q^  =  Qo  +  2Wr)J[C<r]f?T  -f  S^.fcjfdT     .        .  (10) 

InK  =  -  ^   -  ^  fc;<VT  +  ^  f^  </T-,^  ,  +  2r-a 

AX  J.         At  _L  f  "R     I  V  -  , 

»/o  •/ 

/•T  /»T 

where     1       [C^]rfT  -     Ml  dT  is  put  equal  to  zero  (§  201). 
Jo  ^o 

We  thus  arrive  at  the  same  equation  as  for  a  homogeneous  gas 
reaction,  except  that  the  heat  of  reaction  refers  to  all  the  phases, 
gaseous  and  condensed. 

If  we  again  assume,  with  Nernst  (Recent  Applications),  the 
form : 

C;'  =  a,  +  2/8,-T  +  R  =  3-5  +  2&T 
[C'r)]  =  2[£,]T 


THE    THEOREM  OF   NERNST  505 

the  latter  being  certainly  incorrect  at  low  temperatures,  we  find  : 


R  -         P          "o  -  3-5* 

~2f~ 

and  : 

log  K/  =  ~  +WTT  +  l'75v  log  T  +  lift  T  +  S"A   •  <18> 

where  log  K'  =  Sr,  log  7;, 

and  the  other  symbols  have  the  significance  of  §  201.     It  will  be 
observed  that  the  chemical  constants  refer  to  the  gases  only. 
We  may  often  omit  the  term  in  T  and  write  simply  : 

log  K/  =  ~  *OTlT  +  l'75v  log  T  +  B'C<  '  (13) 

and  for  approximate  calculations  may  even  put  : 
Qo  =  Q,,  (approx.) 

when  log  K'  =  -  ^^fc  +  2i'A  .  (14) 

In  connection  with  dissociation  equilibria  of  the  type 

Na2HP04  .  12H20—  NasHPOi  +  12  H20, 
Nernst  points  out  that  the  equation  : 

log  J*  =  ~  -4§fa  +  1-75  log  T  +  C 

where  C  =  3"2  (mean  for  1  mol  of  various  gases,  cf.  §  212) 
leads  to  a  revision  of  the  de  Forcrand  rule  (§  113),  for  when 
p  =  1  atm.  : 

-^  =  4-57  (1-75  log  T  +  3-2) 

which,  according  to  the  rule,  is  approximately  33.  This  can 
only  hold  in  the  middle  of  an  extended  range  of  T. 

It  must  be  borne  in  mind  that  the  equations  (11),  etc.,  hold 
good  only  when  the  condensed  phases  are  chemically  homo- 
geneous. Thus  Foote  and  Smith,  who  determined  the  dissocia- 
tion pressure  of  cuprous  oxide  : 

2Cu20^4Cu  +  0.2 
found  no  agreement  with  Nernst's  equation,  which  is  not  sur- 


506  THERMODYNAMICS 

prising,  as  the  liquid  phase  is  a  solution  of  copper  in  cuprous 
oxide. 

Stahl  (Metallurgie,  1907,  p.  682)  calculated  the  temperatures 
at  which  the  pressures  of  oxygen  over  dissociating  metallic 
oxides  : 

2MOz±2M  +  02 

reach  0'21  atm.  (when  they  are  in  equilibrium  with  atmospheric 
oxygen,  and  begin  to  reduce  freely),  by  means  of  the  equation  : 

log  0-21  =  -  0-6778  =  -        L_  +  1-75  ]og  T  _j_  2-g 


(2-8  is  the  chemical  constant  of  oxygen.) 
He  plotted  log  p  as  abscissae  against  T  as  ordinates,  and  cut 
the  curves  by  an  ordinate  log  p  =  —  0'6778. 

Numerical  Example.  —  Dissociation  of  ammonium  hydrostilphide  : 


Molecular  heat  of  solid       NH^HS  =  19-1 

,,  ,,         gaseous  NH3       =  9'5 

H2S        =  8-5 

Heat  of  dissociation  of  NH4HS        =  22800  cal. 
all  for  ordinary  temperature,  T  =  300 

.-.  Qp  =  21900  +  7-OT  -  0-013T* 

.'.  I  log  K  =  log|  =  -  ^  +  1-75  log  T  -  0-0014T  +  3-15 
(3-15  =  mean  chem.  const.  ==  3'°  +  3'3  for  NH3  and  H2S.) 

For  T  =  298-1,  p  =  0*661  atm.  (obs.).     If  we  putjj  =  0-661  in  the  equation 
we  find  T  =  318.     The  approximate  equation  : 


gives  T  =  312. 

215.     Befthelot's    Rule    and    the    Measurement    of    Chemical 
Affinity. 

If  we  glance  back  over  the  various  branches  of  application  of 
thermodynamics  to  chemical  problems  detailed  in  the  preceding 
sections  of  this  book,  looking  more  especially  at  the  historical 
sequence,  we  shall  find  that  the  physical  chemists  have,  until 
recently,  focussed  their  attention  on  the  theory  of  dilute  solu- 
tions. This  preference  is  due  to  the  great  stimulus  given  by  the 


THE  THEOREM  OF  NERNST        507 

pioneering  work  of  van't  Hoff,  and  th'e  rich  harvest  which  was  then 
thrown  open  to  the  workers  in  the  new  field.  The  older  thermo- 
chemistry, summarised  in  the  famous  "rule"  of  Thomsen  and 
Berthelot  (§  118),  that  the  heat  of  reaction  was  a  measure  of  the 
work  done  by  the  chemical  forces,  was  brought  into  discredit, 
and  violently  attacked  by  the  disciples  of  the  new  school,  and,  so 
far  as  its  theoretical  foundations  went,  was  successfully  deposed. 
At  the  same  time  it  must  have  been  obvious  to  anyone  acquainted 
with  thermochemical  data,  and  not  so  "  purely  chemical  "  as  to 
be  unable  to  carry  out  the  requisite  thermodynamic  calculations, 
that  the  afore-mentioned  rule  was  not  so  useless  as  might  be 
supposed. 

Nernst,  in  his  Theoretische  Chemie,  devoted  a  whole  chapter  to 
a  critical  examination  of  the  rule  of  Thomsen  and  Berthelot,  and 
he  concluded  that  in  many  cases  the  heat  of  reaction  certainly 
does  correspond  very  closely  with  the  maximum  work,  AT, 
which  latter  magnitude  he  took  from  van't  Hoff  as  a  measure 
of  the  chemical  affinity.  AVhilst  pointing  out  that  it  very  often 
gives  results  wholly  incompatible  with  experience,  and  cannot 
therefore  be  indiscriminately  applied,  Nernst  showed  that  the 
rule  nevertheless  holds  good  in  too  many  cases  to  be  wholly  false  ; 
in  an  appropriate  metaphor  he  claimed  that  it  "contains  a  genuine 
kernel  of  truth  which  has  not  yet  been  shelled  from  its  enclosing 
hull."  This  labour  of  emancipation  was  partially  effected  in  the 
newer  work  of  the  same  author,  Application*  of  Thermodynamics 
to  Chemistry,  1907,  which  is  an  attempt  to  place  the  rule  of 
Berthelot  on  a  scientific  basis,  and  to  determine  under  what 
conditions  its  use  is  legitimate.  He  points  out  that  the  equation  : 
AT  =  «,  +  T(Ss  -  SO 

shows  that  it  cannot  be  true  in  changes  involving  a  considerable 
change  of  bound  energy,  nor  those  in  which  substances  of  very 
variable  concentration  participate  (as  in  gas  reactions  and 
reactions  in  dilute  solution).  It  does,  however,  hold  good  in 
the  following  cases : 

(1)  Reactions  between  pure  solids: 

e.g.,  Pb  +  2AgCl  =  PbCla  +  2Ag. 

(2)  Reactions  between  pure  liquids  : 

e.g.,  Pb  +  Br2  =  PbBr2. 


508  THERMODYNAMICS 

(3)  Reactions  between  concentrated  solutions  :  e.g.,  the  electro- 
motive force  of  the  lead  accumulator  corresponds  almost  exactly 
with  the  heat  of  dilution  of  the  acid  when  the  latter  is  concen- 
trated, whilst   in   dilute   solutions   the   difference  is  very  great 
(Nernst,  Wied.  Ann.,  53,  57,  1894). 

(4)  Reactions    occurring    between    dilute    solutions  in   some 
galvanic  cells  (Helmholtz-Thomson  rule,  §  200). 

The  new  theorem  then  introduced  by  Nernst  in  the  form  that 
AT  =  Q  in  the  cases  specified,  and  its  consequences,  have  already 
been  considered. 

Nernst  further  remarked  that  the  stability  of  a  compound  is 
measured  by  the  quantity  which  can  exist  under  given  conditions, 
and  the  previous  equations  enable  one  to  determine  this,  at  least 
approximately,  with  the  help  of  the  heats  of  formation.  A  com- 
pound under  given  conditions  is  in  general  either  very  stable  or 
very  unstable  —  equilibria  in  which  all  the  components  exist 
in  appreciable  concentrations  are  the  exception  rather  than  the 
rule.  A  considerable  heat  of  formation  usually  corresponds  with 
marked  stability  (cf.  HC1,  HBr,  HI). 

Thus,  in  the  formation  of  acetylene  from  carbon  and  hydrogen 
we  have  : 


and  of  benzene : 

3740 


Only  at  very  high  temperatures  would  the  right-hand  side  of 
(a)  have  a  small  positive  value  ;  i.e.,  C2H2  is  formed  under  such 
conditions  (e.g.,  in  the  arc).  Benzene,  however,  must  be  utterly 
unstable  under  all  such  circumstances. 

216.     Application  of   Nernst's  Theorem  to  the  Calculation   of 
Electromotive  Forces:  Condensed  Reactions. 

The  most  simple  application  of  the  equations  : 

[Ar]  =  [<2o]  -  /ST2    .          .          .          .      (1) 

[Q\    =  [<2o]  +  £T2  ....     (2) 

proposed  by  Nernst  (§  209)  for  a  reaction  between  pure   solid 
and   liquid  substances,  is  the  calculation  of  the  electromotive 


THE   THEOREM  OF   NERNST  509 

force  of  a   galvanic    cell  with   the    aforesaid    reaction  as    its 
source  of  current.     Then  : 

[AT]  =  23046E  cal  .....     (3) 
where  E  =  E.M.F.  in  volts. 

Before  Nernst  had  put  forward  his  theory,  Bodlander  (Zeitschr. 
physik.  Chem.,  27,  55,  1898)  had  been  able  to  calculate  the  solu- 
bility of  a  salt  by  the  measurement  of  its  decomposition  voltage, 
and  had  found  that  where  the  reaction  occurring  is  the  dissocia- 
tion of  a  solid  salt  into  solid  uncharged  atoms,  the  work  done  to 
split  up  the  salt  into  its  ions,  and  discharge  these  at  the  electrodes, 
is  very  nearly  equal  to  the  heat  of  formation. 

E.g.,  with  the  cell  Pb/PbI2  sat.  sol./I2(Pt) 

Pb  +  I2  =  Pbl, 

the  E.M.F.  can  be  calculated  thus  : 

(1)  Let  1  mol  PbI2  form  a  saturated  solution  of  concentration 
f.     The  ionic  concentrations  are  £PJ;  =  £/  =  £. 

Now  suppose  this  brought  to  unit  concentration  by  compressing 
with  an  osmotic  piston.  Work  done 

(AT)i  =  RT/n&b  +  2RT/H&'  =  3RT/«£ 

(2)  Discharge  each  ion  at  a  reversible  electrode.     Work  done 
(AT)a  =  sum  of  decomposition  voltages  =  ePb  +  */ 

.-.  total  work  =  (AT)i  +  (AT)2  =  8RTJn£  +  epi;  +  <•/. 
From  the  known  values  of  £,  ePi;,  e/, 


E  can  also  be  calculated  for  the  cell  as  follows  : 

If  [Q]  heat  of  formation  of  1  mol  solid  PbI2  at  T  =  328°, 

2(0  =  19900  cal.  =  heat  of  reaction. 
But  [Q]  =  [<2o]  +  £T2 

.    d2[Q]  _  4pT  _  (initial  heat  capac.)  —  (final  heat  capac.) 

=  cn  +  2Cl  -  cPbl2  =  6-44-2  X  6-86  -  19'7  =  0'4  at  T  =  328° 

4/3  X  328  =  0-4  cal. 
.-.     /S  =0-13  X  10"  '  volt. 


510 


THEKMODYNANICS 


LQ]  =  [<2o]  +  0-13  X  10^7T2 
.-.     E  =  0-864  -0-13  X10~7T2 

=  0-863  volt  when  T  =  328°. 

The  close  agreement  between  0*864  (Thomson  Kule)  and  0'863 
(Nernst's  theorem)  is  also  obvious. 

The  rule  for  the  calculation  of  the  electromotive  force  of  such 
a  cell  is,  therefore,  according  to  Nernst  (cf.  Bed.  Ber.,  1909, 
p.  247)  :  extrapolate  the  thermochemical  data  to  the  lowest 
possible  temperature  and  put  : 

'' 


As  a  further  example  we  may  take  the  cell  : 
Pb  +  2AgCl-*PbCla  +  2Ag 

studied  by  Halla  (Zeitschr.  Elecktrochem.,  14,  411,  1908). 
From  the  thermal  data  of  Breasted  : 

(1)  [  Q  ]  =  11904  +  0-010062T2  -  O'OOOOITIT3 

(2)  .'.    [AT]  =  11904  —  0-010062T2  +  0'00000855T3 


T 

[AT]  obs. 
from  E. 

[Ax]  calc. 
from  (2). 

Diff.  per  cent. 

273-0 

11320 

11328 

+  0-07 

289-7 

11271 

11268 

—  0-03 

303-5 

11220 

11217 

—  0-03 

322-3 

11153 

11144 

-0-08 

331-3 

11123 

11110 

-  0-12 

340-0 

11084 

11077 

—  0-06 

362-0 

10983 

10992 

-0-08 

Finally  we  may  calculate  the  E.M.F.  of  the  Clark  standard 
element.  In  the  ordinary  action  of  this  cell,  the  theorem  is  not 
directly  applicable,  since,  as  Cohen  (Zeitschr.  physik.  Chem.,  34, 
62, 1900 ;  cf.  W.  Jaeger,  Die  Normaldemente,  Halle,  1902)  showed, 
there  is  a  complex  reaction  occurring  in  solution,  the  ZnSOi 
formed  passing  into  the  latter,  combining  with  water,  and  then  on 
account  of  supersaturation  crystallising  out  and  carrying  with 
it  that  portion  of  the  hydrate  dissolved  in  this  water.  But  if 


THE    THEOREM  OF   NERNST  511 

we  cool  down  to  the  cryohydric  point,  when  ice  separates,  the 
reaction  goes  on  between  pure  substances  : 

Zn  +  HgaS04  +  7H20(ice)  =  ZnSO,  .  7H.20  +  2Hg. 
The  heats  of  reaction  are  : 

(Zn,  S,  20a)  =  230090 
(2Hg,  S,  202)  =  175000 
(ZnSO*.  7H-20  liq.)  =  22690 

ice  —  water  =  1580  at  0  =  17 

.-.     2YJ:  =  66720  at  T  =  290 
The  molecular  heats  are  : 
Zn  =  6-0  (10:);    Hg2S04  =  31'0  (50-");     7H20  =  63'7  (10:); 

ZnS04  -  7H20  =  89-4  (10:)  ;  2Hg  =  13"2  UO  ).  The  value  for 
7H20  (ice)  is  extrapolated  from  results  of  Regnault,  Person,  and 
Dewar.  Thus  : 


2         ='  =  4y3T  =  (6  +  31'0  +  63  T)  -  (89'4 

=  —  1-9  for  T  =  288 
.-.  0=—  0-0017 
.-.     [Q\  =  38505  —  0-0017T2 
[AT]  =  38505  +  0-0017T* 

E.M.F.  of  the  Clark  at  the  cryohydric  point 


_  38505  +  0-0017  X  (266)a 

23046 
=  1-4592  volt. 

Pollitzer  (Zeitschr.  Elektrochem,,  17,  5,  1911;  Berechnung 
Chem.  Affinitaten  nacli  Xernstschen  Warmetheorem,  1912)  has 
carried  out  calculations  with  the  new  formula  for  the  specific  heats 
of  solids  (§  204),  but  this  introduces  nothing  new  in  principle. 

217.    Electromotive  Forces  of  Gas  Elements. 

In  §  205  we  have  deduced  the  equation  : 

E  =  ^(log  »!>,«  .  .  .  -  log  K'  - 


512  TH  EEMOLYNAMICS 

for   the   E.M.F.  of   a   gas   element,  where  TTI,  *»-...  are  the 
pressures  of  the  gases  supplied  to  the  electrodes, 
log  K'  =  logjV  pj»  .  .  . 

is  the  equilibrium  constant,  and  v  =  S^i- 

It  is  evident  that  E  is  known  as  soon  as  K'  is  known  for  the 
current-producing  reaction  at  various  temperatures. 

K'  may,  however,  be  calculated  from  Nernst's  theorem  in  the 
manner  explained  in  §  210,  and  hence  the  problem  of  calculating 
the  E.M.F.  of  a  gas  element  is  solved. 

218.     References  to  Chap.  XVII. 

(1)  Griineisen,  Ann.  Phys.  26,  401 ;  C.  L.  Lindemann,  Physik.  Zeitschr  „ 
1.1,  1197,  1912. 

(2)  Bronsted,  Zeitschr.  physik.  Chem.,  55,  371,  1906. 

(3)  Recent  Applications  of  Thermodynamics  to  Chemistry,  Constable,  1907  : 
Theoretische    Chonie,     1    Aufl.,     1913 ;     Pollitzer,    Serechnung  ChemiseJur 
Affinitdten  nach  <hm  Nernstschen  Wdrmetlieortm,  Encke,  1912. 

(4)  J.  H.  van't  Hofif,  Boltzmanns  Festschrift,  1904,  233 ;    J.  N.  Bronsted, 
Zeitschr.  phi/sik.  Chem.  55,  371,  1906;  56,  645,  1906, 

(5)  Schottky,  Zeitschr.  physik.  Chem.,  6%,  433,  1908. 

(6)  0.  Brill,  Ann.  Phys.  [iv.],  21,  170,  1906;  E.  Falck,  Physik.  Zeitschr.  'J, 
433,  1908;  Thiesen,  Verh.  Deutsch.  Physik.  Ges.,  10,  947,  1908. 

(7)  L.  Eollo,  Atti  Accad.  Lincei  [v.],  18,  II.,  365  ;  Chem.  CentralbL,  1910, 
I.,  718;   J.  T.  Barker,  Zeitschr.  physik.  Chem.,  71,  235,  1910. 


CHAPTER  XVIII 

KINETIC    THEOEIBS    IN   THERMODYNAMICS 

219.     Interpretation  of  Thermodynamics. 

Thermodynamics  deals  with  the  various  conditions  under  which 
heat  energy  is  rendered  available  in  physical  and  chemical 
systems.  Energy  changes  are  accompanied  by  definite  changes 
in  the  variables  denning  the  state  of  the  system,  and  a  relation 
established  between  the  quantities  of  energy  leads  to  a  relation 
between  the  properties  of  the  system.  No  detailed  knowledge  of 
the  exact  way  in  which  the  energy  changes  occur  in  the  system, 
is  required,  and  the  methods  of  thermodynamics,  as  contrasted 
with  those  of  the  molecular  theory,  have  the  peculiarity  that 
by  their  aid  we  can  push  forward  the  investigation  of  quantitative 
relations  without  waiting  for  more  intimate  knowledge  of  the 
structure  of  the  system  investigated.  On  the  one  hand  this  is  a 
great  gain,  but  on  the  other  hand  it  is  also  a  decided  loss,  since 
no  information  may  be  obtained  about  that  structure  by  means 
of  thermodynamics  alone. 

For  the  purpose  of  interpretation;  various  hypotheses  have 
been  built  up  around  the  results  which  have  been  derived  from 
the  two  laws  of  thermodynamics.  It  must  not  be  forgotten, 
however,  that  the  deductions  of  thermodynamics  would  stand 
quite  firm  if  the  whole  hypothetical  system  collapsed  about  them. 

The  interpretation  is  usually  mechanical  in  form,  the  various 
energies  being  regarded  as  potential  and  kinetic  energies,  i.e., 
energies  of  configuration  or  motion  of  physical  systems.  The 
acceptance  of  the  atomic  theory  and  the  theory  of  the  ether 
necessarily  involves  the  further  conclusion  that  all  energies  are 
to  be  referred  to  atomic  or  ethereal  configurations  and  motions. 

Potential  energies  then  arise  from  forces  acting  between  the 
parts  of  the  system.  The  nature  of  these  forces  (gravity,  magnetic 
forces,  electrical  forces,  molecular  and  atomic  forces)  is  very 
obscure. 

Kinetic  energies  have  their  seat  in  the  motions  of  the  parts  of 


514  THERMODYNAMICS 

the  system ;  in  this  class  are  supposed  to  reside  heat  energy,  due 
to  molecular  motion ;  the  energy  of  an  electric  current,  due  to 
the  motion  of  electrons,  and  radiant  energy,  due  to  motions  in 
the  ether.  Atomic  energy  is  also  probably  largely  kinetic,  the 
atom  consisting  of  systems  of  negative  electrons  in  motion  in  or 
around  a  positive  charge.  The  electrons  may  be  violently  ejected 
when  the  system  becomes  unstable,  and  the  atom  then  begins 
to  disintegrate,  e.g.,  undergo  radioactive  change. 

The  theory  of  kinetic  energy  is  much  simpler  than  that  of 
potential  energy.  If  the  masses  and  velocities  of  all  parts  of  the 
system  are  known  the  kinetic  energy  is  calculable,  for  it  is  only 
the  sum  of  the  products  of  half  each  mass  into  the  square  of  its 
velocity.  The  specification  of  the  potential  energy,  on  the 
contrary,  involves  a  knowledge  of  the  forces  acting  in  the  system 
and  these  are  usually  complicated  functions  of  the  configuration, 
determinable  only  by  experiment. 

An  explanation  of  potential  energy  involves  an  explanation  of 
force ;  both  terms  are  simply  another  way  of  saying  that  we 
know  nothing  about  the  thing  to  be  explained.  A  distinct 
advance  is  made  when  a  force  can  be  explained  in  terms  of  the 
kinetic  energy  of  a  system  in  motion,  an  illustration  of  which  is 
afforded  by  the  kinetic  theory  of  gases,  which  replaced  the  sup- 
posed forces  of  repulsion  between  the  molecules  of  gases  (the 
existence  of  which  is  disproved  by  Joule's  experiment,  §  73)  by 
molecular  impacts. 

The  relation  between  matter  and  ether  was  rendered  clearer  by 
Lord  Kelvin's  vortex-atom  theory,  which  assumed  that  material 
atoms  are  vortex  rings  in  the  ether.  The  properties  of  electrical 
and  magnetic  systems  have  been  included  by  regarding  the  atom 
as  a  structure  of  electrons,  and  an  electron  as  a  nucleus  of 
permanent  strain  in  the  ether — "  a  place  at  which  the  continuity 
of  the  medium  has  been  broken  and  cemented  together  again 
without  fitting  the  parts,  so  that  there  is  a  residual  strain  all 
round  the  place  "  (Larmor). 

When  these  theories  are  carried  to  their  logical  consequences, 
difficulties  arise,  particularly  in  the  department  of  the  theory  of 
radiation  (cf.  §  224). 

J.  J.  Thomson  and  H.  Hertz  suggested  that  all  energy  may  in 
reality  be  kinetic ;  the  effects  of  force,  and  the  existence  of 
potential  energies  may  then  be  regarded  as  arising  from  the 


KINETIC   THEORIES   IN   THERMODYNAMICS       515 

motion  of  other  concealed  systems  which  are  connected  with  the 
given  system.  As  an  example  we  may  refer  to  the  mechanical 
system  shown  in  Fig.  89  (after  Thomson). 

The  two  balls,  A  and  B,  are  connected,  by  jointed  rods,  with 
the  sliding  pieces  C,  D.  If  the  shaft  XY  is  rotated,  A  and  B 
move  further  apart,  and  C  and  D  approach  each 
other  as  though  attractive  forces  were  operative 
between  them  :  "  This  is  due  to  our  considering  C 
and  D  as  a  complete  system,  whereas  it  is  in  reality 
part  of  a  larger  system,  and  when  we  consider  the 
complete  system  we  see  that  it  behaves  as  if  it 
were  acted  on  by  no  forces  and  possessed  no  energy 
other  than  kinetic." 

In  the  majority  of  cases  the  kinetic  energy  of 
the  concealed  motions  exists  mainly  in  the  ether. 
Thus  the  potential  energy  stored  up  in  separating 
two  unlike  electric  charges  may  be  regarded  as  being  really 
kinetic  energy  due  to  some  kind  of  rotation  about  the  tubes  of 
electric  force  which  exist  in  the  field  around  the  charges. 

220.     Specific  Heats  of  Gases:   Kinetic  Theory. 

The  fundamental  equation  of  the  Kinetic  Theory  of  Gases  is  : 

mu2 
pr  =  -,- 

where  p  =  pressure,  r  =  volume,  m  =  total  mass,  and  n*~  is  the 
mean  square  velocity  of  translation,  of  the  gas  molecules. 
For  unit  mass,  r  =  specific  volume,  and  : 


This  equation  holds  for  an  ideal  gas,  between  the  molecules  of 
which  no  forces  of  attraction  or  repulsion  exist. 
The  kinetic  energy  per  unit  mass  is  : 


or  the  kinetic  energy  per  mol  is  : 


L  L  2 


51fi  THEBMOBYNAMICS 

.-.  the  molecular  heat  at  constant  volume  is  : 
3  3  X  1-985 


The  ratio  of  the  specific  heats  at  constant  pressure  and  at 
constant  volume  is  : 

+  1-985 

These  numerical  values  have  been  verified  for  monatomic  gases. 
If  the  molecule  consists  of  more  than  one  atom,  the  value  of  C(. 
is  greater  than  2'98  and  increases  regularly  with  the  temperature, 
and  the  value  of  K  is  less  than  1'66  and  also  changes  with  the 
temperature. 

The  dependence  of  the  specific  heat  on  the  molecular  com- 
plexity was  explained  by  Boltzmann  (1)  by  the  introduction  of 
energy  of  rotation.  The  value  2*98  is  deduced  on  the  assumption 
that  the  whole  of  the  kinetic  energy  of  the  gas  molecules  is 
translatory.  In  a  monatomic  gas  the  rotational  energy  is  absent, 
or  constant,  for  if  the  centre  of  gravity  of  the  atom  corresponds 
with  the  geometrical  centre  of  the  sphere,  no  change  of  rotational 
energy  can  occur  owing  to  mutual  collisions  when  the  atoms  are 
smooth.  But  in  a  diatomic  gas,  the  two  atoms  may  be  regarded  as 
two  massive  particles  at  a  fixed  distance  from  each  other,  and 
this  dumb-bell  like  structure  can  rotate  in  two  perpendicular 
planes.  Each  degree  of  freedom  of  translatory  or  rotatory 

T>m 

motion  is  supposed  to  have  the  same  energy,  viz.,  -^-,  and  hence 

a 


OTjrn 

a  diatomic  molecule  has  —  -.r-  for  the  translatory  part,  and  —  ^—  for 


the  rotatory  part,  or  —  ~—  in  all.     The  molecular  heat  is  therefore 

Ct.  =  -Q-  =  4*96,  which  is   in  agreement  with   the  values    for 

02,  N2,  etc.,  at  low  temperatures.  That  of  the  halogens  is  greater, 
and  Boltzmann  supposed  the  connexion  between  the  atoms  was 
loosened  with  rise  of  temperature.  Triatomic  gases  have  the 

fiTJ 

value  C.p  =  —fr  =  5  '95,  which  is  exhibited  by  steam  and  carbon 

Zi 

dioxide  at  low  temperatures.'2' 


KINETIC   THEORIES  IN   THERMODYNAMICS       517 

221.     Specific  Heats  of  Solids:   Kinetic  Theory. 

Boltzmann  <:5)  extended  the  theory  to  solids,  and  was  led  to  a 
result  which  to  a  certain  extent  is  in  harmony  with  the  law 
of  D  along  and  Petit. 

An  elementary  solid,  such  as  silver,  is  regarded  as  composed 
of  atoms  oscillating  about  fixed  centres.  The  total  energy  content 
is  therefore  partly  kinetic  and  partly  potential.  Since  the  solid 
has  a  finite  compressibility,  the  atoms  may  be  supposed  to  be 
maintained  at  small  distances  apart  by  forces  they  exert  upon 
one  another,  and  these  may  be  resolved  into  two  sets,  one  of 
which  opposes  a  closer  approximation  of  the  atoms,  and  the  other 
tends  to  draw  the  latter  together.  Both  are  functions  of  the 
distance  between  the  atoms,  and  for  a  given  distance  are  equal, 
since  the  form  of  the  body  is  altered  by  external  forces  alone. 

The  force  exerted  on  an  atom  in  its  position  of  rest  by  the 
neighbouring  atoms  is  then  : 

</>(,-)  -  </>(>•)  =  0. 

The  two  functions  have  the  same  form,  since  any  atom  may 
be  selected,  and  the  system  is  symmetrical. 

If  the  atom  is  displaced  from  its  equilibrium  position  through 
the  distance  Sr,  the  force  of  restitution  is  : 

ftr  +  Sr)  -  <f>(r  -  6>)  =  </>(r)  +  </>'('0^  +      (}  (Sr)2  +  .  .  . 


At  low  temperatures  the  vibrations  are  small 

/.  elastic  force  =  %<f>'(r)8r  =  a,  say, 
and  the  dynamical  equation  of  motion  is  : 


where  m  =  mass,  x  =  displacement,  t  =  time. 
The  solution  of  the  equation  is : 

x  =  A  cos  (kt  +  e) 


where  k  = 


518  THERMODYNAMICS 

i.e.,  the  particle  executes  simple  harmonic  motions  of  frequency  : 

-  L       JL     /?> 

~  2w  ~~  2^  V  „/ 

The  velocity  is  -rr  =  —  A/c  sin  kti  at  the  instant  /  =  ti 

A2a 
.*.  the  kinetic  energy  per  unit  mass  is  T  =  -=-  sin  akti. 

To  calculate  the  potential  energy  we  assume  an  expression  of 
the  form  : 


If  x  is  small,  all  terms  after  ex*  may  be  neglected.     Also  Y 
must  be  a  minimum  in  the  equilibrium  state  x  =  0 


and  since  it  vanishes  in  the  equilibrium  state,  a  is  also  zero, 

.-.  V  =  ex* 
But  T  +  V  =  const. 

since  no  external  forces  are  acting  on  the  system, 

i  (dx\  2   . 
.*.  \  I  -T-  )    +  cx2  =  const. 

dx     d*x    .  dx 


°r         C=-  =  - 


The  mean  values  of  cos  kti  and  sin  /cfi  over  any  interval  of  time 
including  a  whole  number  of  vibrations  are  each  ^, 

.  T  =  V  =  —  . 
4 

The  mean  values  of  the  kinetic  and  potential  energies  of  an 
atom  are  therefore  equal  over  any  interval  of  time,  and  if  we  add 
together  all  the  kinetic  energies  and  all  the  potential  energies  of 


a 


KINETIC   THEORIES   IN   THERMODYNAMICS      519 

a  large  number  of  atoms  in  a  solid,  which  are  vibrating  in 
the  manner  described,  the  two  sums  will,  at  every  instant,  be 
equal. 

Now  let  the  silver  be  supposed  to  be  surrounded  by 
nionatomic  gas,  such  as  helium.  The  atoms  of  the  gas,  when 
they  collide  with  those  of  the  solid,  will  impart  kinetic  energy  to 
them.  According  to  Maxwell's  Law  of  Equipartition  of  Energy, 
two  bodies  are  in  temperature  equilibrium  when  the  mean  kinetic 
energy  for  each  degree  of  freedom  of  motion  of  each  atom  is  the 
same  in  both.  The  atoms  of  the  solid  and  those  of  the  gas  have  three 
degrees  of  freedom  each  ;  to  each  degree  corresponds  the  kinetic 

RT  QT?T 

energy  -^-,  therefore  the  kinetic  energy  of   the  solid  is  -—  . 

The  potential  energy  is  equal  to  this,  and  the  total  energy  content 
is  therefore,  per  mol 

U  =  3RT 

.-.  C,  =  (?U/aT)c  =  3R  =  5-955 

which  agrees  moderately  well  with  the  law  of  Dulong  and  Petit. 

Abnormally  low  atomic  heats  were  explained  by  Richarz  on  the 
assumption  of  a  diminution  of  freedom  of  oscillation  consequent 
on  a  closer  approximation  of  the  atoms,  which  may  even  give 
rise  to  the  formation  of  complexes.  This  is  in  agreement  with  the 
small  atomic  volume  of  such  elements,  and  with  the  increase  of 
atomic  heat  with  rise  of  temperature  to  a  limiting  value  6'4,  and 
the  effect  of  propinquity  is  seen  in  the  fact  that  the  molecular 
heat  of  a  solid  compound  is  usually  slightly  less  than  the  sum  of 
the  atomic  heats  of  the  elements,  and  the  increase  of  specific  heat 
with  the  specific  volume  when  an  element  exists  in  different 
allotropic  forms. 

There  is,  however,  a  fatal  objection  to  the  theory  of  Boltzmann. 
At  very  low  temperatures  the  oscillations  will  be  small,  and 
should  conform  to  the  theory.  But  the  atomic  heats,  instead  of 
approaching  the  limit  5'955  at  low  temperatures,  diminish  very 
rapidly,  and  in  the  case  of  diamond  the  specific  heat  is  already 
inappreciable  at  the  temperature  of  liquid  air.  A  new  point  of 
view  is  therefore  called  for,  and  it  is  a  priori  very  probable  that 
this  will  consist  of  a  replacement  of  the  hypothesis  of  Equiparti- 
tion of  Energy  adopted  by  Boltzmann.  This  supposition  has 
been  verified,  and  the  new  law  of  partition  of  energy  derived 


520  THERMODYNAMICS 

from  investigations  in  quite  another  department  of  physics,  viz., 
in  the  theory  of  radiation. 

222.     Planck's  Theory  of  Radiation. 

In  his  theoretical  investigation  on  the  exchange  of  energy 
between  ether  and  matter,  Planck  (4)  considered  the.  absorption 
and  emission  of  radiation  by  a  linear  electrodynamic  resonator, 
i.e.,  a  system  composed  of  two  opposite  electric  charges,  oscillating 
on  a  fixed  straight  line.  If  a  closed  space  contains  a  number  of 
such  resonators,  of  small  damping,  and  at  large  distances  apart, 
the  effect  of  an  exciting  electromagnetic  radiation  of  definite 
colour  on  the  system  may  be  considered.  Such  radiation,  in  so 
far  as  it  is  open  to  observation,  must  not  be  regarded  as  com- 
posed of  a  single  vibration  of  absolutely  definite  frequency,  but  as 
covering  a  definite  breadth  of  the  range  of  frequencies,  i.e.,  it 
would  appear  in  the  spectroscope  as  an  extremely  narrow  band, 
not  as  a  line.  The  law  of  Kirchhoff,  that  in  a  space  containing 
bodies  of  any  character  whatever,  the  distribution  of  radiation 
ultimately  settles  down  into  a  steady  state  of  thermodynamic 
equilibrium,  is  interpreted  by  Planck  as  indicating  that  in  such  a 
space  there  exists  a  magnitude  which  always  increases  in  temporal 
changes,  and  attains  a  maximum  in  thermodynamic  equilibrium ; 
this  is  the  electromagnetic  entropy  (cf.  §  48  (1) ).  The  question  as 
to  the  distribution  of  frequencies  over  a  small  range  in  a  radiation 
of  definite  "  colour  "  is  solved  on  the  assumption  that  in  natural 
radiation  the  deviations  of  single  rapidly  varying  magnitudes 
from  their  mean  value  shall  be  random,  and  the  application  of 
the  theory  of  probabilities  is  carried  over  from  the  kinetic  theory 
of  gases  to  the  theory  of  radiation. 

The  energy  per  unit  volume,  and  the  entropy,  of  radiation  in 
equilibrium  with  a  system  of  resonators  of  frequency  v  can  then 
be  calculated. 

The  remarkable  result  of  this  investigation  which  is  of  interest 
to  us  here,  is  the  hypothesis  introduced  by  Planck  that  the 
energy  of  a  resonator  does  not  increase  continuously,  like  the 
kinetic  energy  of  a  particle  moving  in  a  straight  line  under  the 
action  of  a  force,  but  per  saltum,  in  whole  multiples  of  a  quantity 
€,  proportional  to  the  frequency : 

<  =     /3* (1) 


KINETIC   THEORIES  IN   THERMODYNAMICS      521 

where     R  =  gas  constant  =  8'315  X  107  r^- 

N  =  number  of  molecules  in  a  mol  =  60  X  10'22 
/3  =  a  universal  constant  =  4'86  X  10  ~  n 

.-.  e  =  6-551  X  v  =  h  v  .         .         .         .     (2) 

e  is  called  the  Energiequantuin  for  the  frequency  v  ;  we  may 
refer  to  it  briefly  as  the  ergon. 

Einstein  <5>  remarked  that  this  point  of  view  can  be  carried 
over  to  the  theory  of  the  energy  content  of  a  solid  body  if  we 
suppose  that  the  positive  ions  of  Drude's  theory  (§  198)  may  be 
looked  upon  as  the  vibrating  resonators,  and  the  seat  of  the 
heat  content  of  the  body  (KSryerwarm^),  He  calculated  the 
expression : 

U-    3K^  (3) 

U   —    -£— •  .  .  .  .      (6) 

t'T   -  1 

for  the  atomic  heat  of  a  solid  composed  of  such  spacial  resonators, 
each  equivalent  to  three  linear  electrodynamic  resonators,  with 
one  definite  frequency  v.  The  atomic  heat  is  therefore : 

cr  =  au/?T 


(4) 


It  may  be  remarked  that  there  is  no  call  for  an  atomic  theory 
of  energy,  analogous  to  the  atomic  theories  of  matter  and  elec- 
tricity, as  the  discontinuity  arises  from  the  peculiar  character 
of  the  system  (cf.  Planck,  Bet:,  45,  5,  1912). 

223.     Specific  Heats  of  Solids:    Einstein's  Equation. 

We  consider  again  a  monatomic  solid  in  contact  with  a  mon- 
atomic  gas  (§  221). 

The  atoms  of  the  gas,  by  collision  with  those  of  the  solid, 
give  up  energy  to  them,  and  we  have  to  find  the  way  in  which 
the  energy  of  the  system  is  distributed  between  the  gas  and  the 
solid  when  there  is  equilibrium. 

For  the  distribution  of  velocities  in  the  gas,  in  any  given  plane 
we  have,  according  to  Maxwell's  distribution  law  : 

•»  +  * 

dN  =  N0A'2<>       *~~dudr  .          .         .     (1) 


522 


THERMODYNAMICS 


where  (IN  denotes  the  number  of  molecules  in  the  total  number 
NO,  the  velocities  of  which  lie  between  u  +  du  and  r  +  dv.     If 

we  transform  to  polar  co-ordinates,  and  put  — — -  =  E   for    the 
kinetic  energy  of  a  particle  m,  we  find : 

dN  =  N0A<?       dE   .        '.',.'       .     (2) 

We  now  assume  that  the  energies  equalise  in  the  plane,  then 
if  we  suppose  the  N0  atoms  of  the  solid  arranged  according  to 
their  energy  content  along  the  axis  of  abscissae,  and  the  corre- 
sponding energies  taken  as  ordinates,  we  obtain  the  continuous 


FIG.  90. 

curve  of  Fig.  90,  on  the  assumption  that  the  exchange  of  energy 
occurs  continuously. 

The  area  under  the  curve  represents  the  total  energy,  RT- 
Now: 

•N  /*E  E 


I  rfN  =  N  — 

Jn  Jf 


=  N  —     N0A< 


Eo 


/E  =  (Ai— 


But  when  E  =  0,  N  =  0,and  when  E  =  oo  ,  N  =  N0  /.  AI  =  A2=  1 
.-.  N==N0l -«" 


or 


KINETIC   THEORIES   IN   THERMODYNAMICS      523 

J"NO 
ErfN  =  E0N0  =  RT 
j 

.•.E0=«T        .        .        .        .     (8) 

is  the  mean  energy  of  an  atom  in  a  given  degree  of  freedom. 

If  we  now  suppose  that  an  atom  of  the  solid  can,  by  collision 
with  a  gas  atom,  take  up  only  a  whole  number  of  ergons  of  the 

T> 

magnitude  e  =  ^-  j3v,  the  energy  content  is  given  by  the  area 

i>o 

under  the  step-formed  curve,  which  shows  that  a  certain  number 
of  atoms  are  at  rest,  with  zero  energy,  another  set  are  all  vibrating 
with  the  energy  e,  corresponding  with  the  number  of  gas  atoms 
the  kinetic  energy  of  which  increases  continuously  from  e  to  2e, 
and  so  on.  The  larger  the  ergon,  the  less  will  be  the  total 
energy  content  of  the  solid  in  comparison  with  the  energy 
of  the  gas  atoms  ;  when  the  ergon  is  small  (e.g.,  with  lead) 
the  two  areas  become  nearly  equal.  If  we  multiply  by  3 
to  include  the  three  degrees  of  freedom,  we  have  for  the  total 
energy  content  : 

"I 


Ut  2e\  /        2<r  3e\ 

c  ~  EO  -,'  ^Foj  +  2  ^T*  -  c  -*)  +  3  ^   .  . 


_    e  _2e  _3f_ 

=  3eN0   e    NO  +  e    S  +  e    NO  +  .  . 


.MO  _  1 


(4) 


Thence  C,  =  0U/3T  =  8E      --       .        .        .        .     (5) 


which  is  Einstein's  equation.  The  present  deduction  is  due  to 
Nernst.(2)  It  leads  to  an  entirely  novel  conception  of  the  con- 
dition of  a  solid  body  at  low  temperatures.  Since  the  energy 
content  must  vanish  for  T  =  0  (in  agreement  with  the  experi- 
mental result  that  [Cp]T=io=  0),  we  must,  at  low  temperatures, 
have  only  here  and  there  a  vibrating  atom,  enclosed  in  a  matrix 
of  atoms  at  rest.  At  absolute  zero,  all  the  atoms  rest  in  their 
positions  of  equilibrium.  This  view  of  the  state  in  which 


524  THERMODYNAMICS 

the  energy  of  a  solid  is  divided  among  the  atoms,  we  shall  call 
briefly  the  Theory  of  Ergonic  Distribution,  in  contrast  with 
the  older  theory  of  Boltzmann,  according  to  which  every  atom 
vibrates,  and  the  energies  of  the  atoms  are  distributed  according 
to  Maxwell's  law  of  equipartition  of  energy. 

224.     Testing  the  Theory  of  Ergonic  Distribution. 

Towards  the  experimental  verification  of  the  theory  of  ergonic 
distribution,  five  lines  of  investigation  have  been  struck  out. 

(1)  Theory  oj  Radiation. 

For  radiation  in  equilibrium  with  electromagnetic  oscillators 
consisting  of  the  charges  on  material  ions  we  can  combine  the 
formula  for  the  mean  energy  of  a  resonator  (Planck,  Wanne- 
strahlnna,  p.  124) : 


(where  c  =  velocity  of  light  in  vacuum,  and  />„  is  the  volume 
density  of  the  radiant  energy),  with  the  expression  for  the  mean 
energy  of  the  atom. 

For  the  latter  we  may  take  either  Boltzmann 's  expression  : 

U   =—  (2a) 

No 
or  that  of  Einstein  : 

.        .   •     .     (26) 
1 
and  arrive  at  the  equation  of  Rayleigh  : 


.      .      .      .   (8a) 

or  of  Planck : 

R        8*1?  0I> 

'•^No"?"   -*r-  .  '         '     (3b) 

e    T  —  1 

respectively. 

With  regard  to  these  we  may  simply  quote  a  remark  of  Lorentz 
(Theory  of  Electrons,  Leipzig,  1909,  p.  287) :  "  The  only  equation 
by  which  the  observed  phenomena  are  satisfactorily  accounted 
for  is  that  of  Planck,  and  it  seems  necessary  to  imagine  that,  for 
short  waves,  the  connecting  link  between  matter  and  ether  is 


KINETIC   THEORIES  IN   THERMODYNAMICS      525 

formed,  not  by  free  electrons,  but  by  a  different  kind  of  particles, 
like  Planck's  resonators,  to  which,  for  some  reason,  the  theory  of 
equipartition  does  not  apply.  Probably  these  particles  must  be 
such  that  their  vibrations  and  the  effects  produced  by  them 
cannot  be  appropriately  described  by  means  of  the  ordinary 
equations  of  the  theory  of  electrons ;  some  new  assumption,  like 


40  60 

Aba,  Temperature 

FIG.  91. 


Planck's  hypothesis  of  finite  elements  of  energy  will  have  to  be 
made." 

(Cf.   Jeans,   Phil.   Mag.    [vi.],   10,   91,   1905  ;     Planck,   Acht 
Forlesungen,  p.  95  ;  Ann.  d.  Phys.,  1912.) 

From  his  equation  Planck  was  able  to  calculate  the  number  of 
atoms  in  a  gram-molecule : 

No  =  61-75  X  1022 


526  THERMODYNAMICS 

which  is  in  excellent  agreement  with  the  values  deduced  in  other 
quite  different  ways. 

(2)  Direct  Measurements  of  Specific  Heats  at  Low  Temperatures. 

By  means  of  the  experimental  methods  briefly  referred  to  in 
§  9  a  large  number  of  specific-heat  measurements  have  been 
made  at  very  low  temperatures.  In  Fig.  91  we  have  the  atomic 
heats  of  some  metals,  and  of  the  diamond,  represented  as  functions 
of  the  temperature.  The  peculiar  shape  of  the  curves  will  be  at 
once  apparent.  At  a  more  or  less  low  temperature,  the  atomic 
heat  decreases  with  extraordinary  rapidity,  then  apparently 
approaches  tangentially  the  value  zero  in  the  vicinity  of  T  =  0. 
The  thin  curves  represent  the  atomic  heats  calculated  from  the 
equation : 


-  a,  =  3R 


with  the  values  of  fiv  indicated  alongside. 

Since  the  specific  heat  actually  measured  is  C,,  the  value  of 
Ctf  must  bs  calculated  from  the  equation  (3)  of  §  64  : 

m  9*'      dp 
TW    8T 

Griineisen  (Ann.  Phys.,  26,  401)  gives  the  expression  : 


where   a  =  coefficient  of  linear  expansion 

f]  =         „         ,,  compressibility 

a  =  atomic  volume. 

There  is,  however,  little  exact  data  for  its  application. 
Lindemann  and  Magnus  (6)  found  that  C,,  could  be  fairly  well 
represented  by  adding  to  Einstein's  equation  an  arbitrary  term 

aT^  where  a  =  const. 


In  the  case  of  ice,  the  term  aT6  was  used. 


KINETIC   THEORIES   IN   THERMODYNAMICS       527 

In  the  case  of  compounds,  the  terms  for  the  separate  atoms 
must  be  added  together,  and  : 


where  the  summation  extends  over  all  values  of  v. 

The  same  equation  may  be  used  when  an  element  has  more 
than  one  value  of  v.  At  very  low  temperatures  this  correction 
term  is  negligible  ;  according  to  Lindemann  and  Magnus,  it  takes 
account  of  the  work  done  against  the  force  of  cohesion  during 
the  expansion.  The  influence  of  the  heat  capacity  of  the  free 
electrons  which  on  Drude's  theory,  together  with  the  positive 
ions  bearing  the  internal  vibrational  heat  energy,  compose  the 
solid,  appears  to  be  negligibly  small,  since  the  atomic  heats  at 
very  low  temperatures  (when  the  part  due  to  the  positive  ions 
considered  in  Einstein's  theory  is  very  small)  are  exceedingly 
small  (cf.  Nernst,  BerL  Ber.,  12,  247);  and  in  addition  the 
atomic  heat  of  lead,  with  its  small  ergon,  exceeds  only  very 
slightly  the  value  3R.  Measurements  at  the  low  temperature 
of  liquid  hydrogen  (Nernst,  loc.  cit.)  show  that  the  curves  do  not 
follow  the  Einstein  equation  exactly  ;  the  reason  is  not  yet  clear, 
but  Lindemann  (Diss.,  p.  38)  suggests  a  polymerisation  of  some 
20  atoms  to  a  molecule,  or  that  some  atoms  are  rigidly  connected 
together,  and  vibrate  more  slowly.  In  the  case  of  sylvine  (KC1) 
the  observed  curve  lies  above,  but  almost  parallel  with  the 
Einstein  curve,  and  Nernst  and  Lindemann  (7)  used  the  equation  : 

r  -3R2     gT\T7 

T  '7~^iV  + 

[>T-v 

which  represents  the  relations  quite  well. 

The  energy  per  mol  according  to  the  new  formula  is : 


U  =  |R 
i.e.,   consists   of  two    parts   which  approach    equality   at   high 


528  THERMODYNAMICS 

temperatures.  According  to  Boltzmann's  theory,  the  energy  of 
a  solid  is  half  kinetic  and  half  potential ;  with  the  new  assumption 
this  cannot  be  true  except  at  high  temperatures,  when  in  fact  the 
deductions  from  the  theory  of  ergonic  distribution  pass  into 
those  from  the  older  theory  (cf.  §  267). 

(3)  Melting -Point  and  Atomic  Volume :  Law  of  Dulong  and 
Petit. 

Lindemann  (8)  has  made  an  interesting  application  of  the 
new  theory  in  the  determination  of  the  frequency  of  atomic 
vibration,  v,  from  the  melting-point.  He  assumes  that  at  the 
melting-point,  T,,  the  atoms  perform  vibrations  of  such  amplitude 
that  they  mutually  collide,  and  then  transfer  kinetic  energy  like 
the  molecules  of  a  gas.  The  mean  kinetic  energy  of  the  atom 
will  then  increase  by  fRT,,  when  the  liquid  is  unpolymerised 
and  the  fusion  occurs  at  constant  volume  ;  this  is  the  molecular 
heat  of  fusion. 

Let  r  =  distance  between  centres  of  two  neighbouring  atoms, 

p  =  distance  between  the  surfaces  as  a  fraction  of  r, 
then  the  diameter  of  the  sphere  of  molecular  activity  is 

<r  =  r(l-/>)       •  •  •  •        (1) 

and  -£    =  distance  through  which  an  atom  must  swing  from  its 

a 

equilibrium  position  to  hit  its  neighbour. 

The  kinetic  energy  when  passing  its  position  of  equilibrium  is 
therefore : 


- 


(2) 


where  a  —  2<£'(V)  i8   ^e   quasi-elastic    force   opposing   atomic 
approximation  (§  221). 

But,  according  to  Einstein's  equation,  the  kinetic  energy  of  an 
atom  in  its  equilibrium  position  at  the  commencement  of  fusion 
is,  with  assumed  linear  vibrations : 
.T.,  /  /Q,,\  2    pv 

r 


r  • 
_R_ 

"NO 

J  n 


>*T  =  I 


,*_lJ  eT>  -1 


=  |0(T.-^)approx.    .         .     (8) 


KINETIC   THEORIES   IN   THERMODYNAMICS      529 

Thus 


or 


which  gives  the  value  of  the  force. 
Thence  : 


'  ~  27r  V   ^~  2^ 


(5) 


P2/-2N0w 

where  m  =  mass  of  an  atom  /.     N0m  =  atomic  weight  =  M. 

If  p  is  assumed  alike  for  all  solids,  and  the  second  term  is 
neglected,  we  find : 


where  V  =  atomic  volume. 

The  values  of  "  v  obs."  were  determined  from  the  atomic  heats 
by  means  of  Einstein's  formula,  those  of  "  v  calc."  were  obtained 
from  equation  (6) : 


T* 

vX  10  -  12  obs. 

v  X  10-  l-  calc. 

Pb" 

600 

1-44 

1-4 

Ag 

1234 

3-3 

(3-3) 

Cu 

1357 

4-93 

5-1 

I 

386 

1-5 

1-4 

Pt 

2018 

31 

3-1 

Si 

1703 

10-7 

7"2 

(A-  =  2'12  X  1012  from  the  atomic  heat  of  silver.) 

The  deviations  from  the  law  of  Dulong  and  Petit  may  therefore 
be  quantitatively  calculated,  at  any  temperature,  from  the  melting- 
point  and  atomic  volume.  Elements  with  high  melting-point 
and  small  atomic  volume  (e.g.,  carbon)  deviate  from  the  law,  the 
value  of  v  being  all  the  greater  the  higher  the  melting-point  and 
density,  and  the  smaller  the  atomic  weight.  In  the  case  of 
lithium,  the  effect  of  the  small  atomic  weight  is  compensated  by 
the  low  melting-point,  and  the  atomic  heat  is  normal. 


530  THERMODYNAMICS 

According  to  Joule's  law  (§  9),  the  molecular  heat  of  a 
compound  is  the  sum  of  the  atomic  heats  of  its  components,  and 
since  this  holds  good  even  when  the  atomic  heats  are  "  irregular," 
i.e.,  not  equal  to  6'4,  it  seems  that  the  heat  content  of  a  solid 
resides  in  its  atoms,  and  not  in  the  molecular  complexes  as  such. 
This  agrees  with  Einstein's  theory.  Hence  the  molecular  heat 
of  a  compound  should  be  calculable  by  means  of  the  formula  : 


-  I 

from  the  atomic  heats  of  its  constituents  in  that  one  extends  the 
summation  over  the  ^-values  of  the  latter.  The  frequency  of 
an  atom,  should  therefore  be  the  same  in  the  free  state  and  in 
combination. 

That  this  is  not  the  case  follows  from  the  experimental  data 
discussed  by  A.  Russell  (9),  and  F.  Koref  (10)  has  attempted  to 
calculate  the  change  of  frequency  of  an  element  when  it  enters 
into  combination  by  means  of  the  alteration  of  melting-point 
and  atomic  volume.  According  to  Lindemann's  equation,  for  the 
combined  atom  : 


For  two  atoms  in  a  binary  compound 
;B   3  AV  _  rb 

where  ra,  rb  are  the  atomic  radii. 
For  the  free  elementary  atom  ; 


v'  /T  '    s   /V 

'"'  ^  =  V~T,  VT 


If  the  molecular  volume  is  the  sum  of  the  atomic  volumes 
V'  =  V 


•    ^-   /T? 

'  *  "  V  T; 


KINETIC   THEORIES  IN   THERMODYNAMICS      581 

The  change  of  melting-point  can  produce  a  very  marked 
alteration  of  v  (e.g.,  carbon  and  carbon  dioxide).  Koref  calculates 
the  v'  values  for  a  number  of  compounds,  and  the  molecular 
heats  thence  obtained  agree  with  the  experimental  values 
moderately  well. 

(4)   The  Theorem  of  Nernst. 

Nenist  <n)  himself  made  a  suggestion  towards  a  kinetic  inter- 
pretation of  his  theorem  : 

Lim     '?A  _  n 

T  =  o      (/T  - 

"  Since  at  the  absolute  zero  the  kinetic  energy  is  zero,  the 
maximum  work  is  the  sum  of  the  differences  between  the  potential 
energies  of  all  the  atoms  before  and  after  the  reaction.  By  the 
motion  of  the  atoms  which  corresponds  to  a  definite  elevation  of 
the  temperature  above  absolute  zero  these  potential  energies  are 
evidently  changed.  The  above  equation  requires  that  this  change 
shall  be  either  infinitely  small,  or  independent  of  the  state  in 
which  the  atom  exists." 

It  is  evident  when  we  express  the  theorem  in  the  form  : 


that  the  interpretation  is  coextensive  with  a  kinetic  interpretation 
of  the  entropy.  According  to  Boltzmann  (§  50)  the  latter 
may  be  regarded  as  proportional  to  the  logarithm  of  a  certain 
probability  of  the  occurrence  of  a  stable  state  characterised  by 
the  most  random  distribution  (elemcntar  Unordnung)  of  the 
elements  composing  the  system.  When  this  chaos,  or  "  mixed- 
upuess  "  (Gibbs)  is  as  great  as  possible,  the  entropy  is  a  maximum. 
Thus,  the  latter  occurs  when  two  gases  are  uniformly  mixed.  When 
the  entropy  vanishes,  the  disorder  must  also  be  a  minimum,  and 
this  leads  at  once  to  the  kinetic  interpretation  of  the  vanishing 
specific  heat.  The  vibrations  cannot  be  of  the  kind  postulated 
by  Boltzmann,  in  which  every  atom  possesses  some  kinetic 
energy  (except  at  the  absolute  zero),  and  the  kinetic  energies 
are  distributed  according  to  the  probability  law  expressing  a  state 
of  elementary  chaos  ;  rather,  there  must  be  a  certain  amount  of 
order,  and  this  corresponds  with  the  ergonic  distribution,  where 
a  fraction  only  of  the  atoms  are  vibrating  with  definite  energies. 
The  theorem  has  been  discussed  by  F.  Jiittner,  who  starts  with 

M    M    2 


532  THERMODYNAMICS 

the  expression   for  the  entropy  deduced   by  Planck   from  the 
statistical  theory : 


...  [U3  =  8B^       .        ,       .        .    (3) 
in  agreement  with  §  223  ; 

and[U]T  =  o  =  0        .    "    .         .         .     (5) 

i.e.,  the  vibrational  energy  vanishes  at  T  =  0.  There  may  be  a 
potential  energy  at  T  =  0,  due  to  chemical  composition,  and  if  we 
call  this  [B0]  we  have  for  the  total  energy  at  any  temperature : 

[E]  =  [E0]  +  [U] 
The  free  energy  is  : 

[*]  =  [E]  -  T[S]  =  [E0]  -  8R/&;  +  3RT/»  ($-  l)  .     (6) 


For  a  reaction  2na  =  0  between  condensed  substances  with 
evolution  of  heat  [Q],  we  have  for  the  maximum  work,  i.e.,  the 
loss  of  free  energy  : 

[A,]  =  [Q]  -  TA[S]  .         .         .         .     (7) 


[Q]  =-  [0o]  ~  3RS»   j-      ;  .         .         .     (8) 
CT_I 

and  A[S]  is  (4)  multiplied  by  Sw 

.%  [AT]  =  [Q0]  -  8BSw08i')  +  8BTSw/w    ,•¥_  l         .     (9) 


and  -         ==  2»[C(.]  =  3RS»        -  ,         -     (10) 


thus  from  v  we  can  find  [<2j,  [AT]  and  A[S]  for  all  temperatures 
if  we  know  [Q]  for  one. 


KINETIC   THEORIES   IN   THERMODYNAMICS      533 

Particular  interest  attaches  to  the  values  assumed  by  [Q]  and 
[AT]  near  absolute  zero.     If  we  expand  (4)  we  find 

.        .        .        .    (11) 


for  small  values  of  T.  If  we  now  differentiate  repeatedly  with 
respect  to  T,  and  retain  only  those  terms  which  are  important  for 
small  values  of  T  we  find  : 

</"[S]  _    3R       (J3»r  +  1 
(IT11  ~  T2"  -  l  '        * 

The  denominator  is  infinite  for  n  =  1,  2,  3,  .  .  .  ,  and  T  =  0, 


..=      Lim  =  0   .        .        .     (12) 

T=  +o      WT"   / 

a  remarkable  property  of  the  entropy. 

From  (6)  and  (11)  we  have  for  small  values  of  T  : 

[*]  =  ;EO]  +  BR  ^  -  3RT  ^  =  :E0;  approx.     .     (13) 


and  thus  Nernst's  theorem  is  closely  related  to  the  theory  of 
Einstein.     If  we  now  differentiate  [*]  repeatedly  we  find  : 


<7T"  r/T"  ~  ' 

f/'*-1 

Lini      -^  =  Lim 

T=  +  o      '/T  r=+o 

r/2^! 

Lim       j~~  =  Lim 

T  =  -t-  0        "A  T  =  -r  0 


^"[*1  T .        « 

Lim      -—i1  =        Lim      -i^-- 

T  =  +  o         "T  T  =  +  0        «1 

i.e.,  the  order  of  contact  of  the  curves  [Q]  and  [AT]  is  infinite  at 
T  =  0  i  e.f  they  coalesce  for  small  values  of  T.     Also : 


534  THERMODYNAMICS 

i.e.,  not  only  the  specific  heat,  but  all  its  temperature  coefficients, 
vanish  at  T  =  0. 

The  relations  can  be  carried  over  to  a  chemical  reaction  if  we 
multiply  by  ^n,  and  put  : 

S»[E]  =  —  [Q]  the  heat  of  reaction, 
Swf*]  =  —  [AT]  the  maximum  work. 

The  expression  : 


shows  that  there  is  a  singular  point  at  T  =  0  ;  for 

Lim  [E]  =  [E0] 

T  =  +  0 

but  Lim  [E]  =  [E0]  +  SRfiv 

T  =  -  0 

i.e.,  [E],  and  therefore  [Q],  can  be  calculated  above  a  temperature 
T,  for  which  it  is  known,  but  not  below  it,  by  means  of  a 
Maclaurin's  series  : 


which  supposes  [E]  uniform  at  T  =  0. 

The  correct  expression  is  a  Laurent's  series,  with  integral 
negative  powers  of  T. 

225.     Rotational  Energy. 

According  to  Nernst  (2)  the  property  of  taking  up  energy  in 
definite  discrete  quantities  belongs  also  to  systems  of  masses  in 
rotation,  in  particular  to  gas  molecules,  composed  of  two  or  more 
atoms,  which  are  rotating  in  the  sense  of  Boltzmann's  theory 
(§  220).  The  ergon  in  this  case  is  assumed  to  be  proportional 
to  the  velocity,  which  in  turn  is  proportional  to  the  square  root 
of  the  temperature.  Thence,  the  molecular  heat  of  an  n-atomic 
gas  at  constant  volume  is  an  expression  composed  of  three  terms. 
The  first  term,  fR,  belongs  to  the  kinetic  energy  of  translator}7 
motion  of  the  molecule  en  bloc,-  the  second  to  the  rotational 
energy,  the  sum  being  taken  over  three  mutually  perpendicular 
planes  ;  and  the  third  to  the  vibrational  energy  of  the  atoms 
about  their  centres  of  equilibrium  in  the  molecule,  the  sum  being 
taken  over  all  p-values  and  planes  of  vibration. 


KINETIC   THEORIES   IN   THERMODYNAMICS      535 

A  very  interesting  calculation  has  been  carried  out  by  N. 
Bjerrum  (Zeitschr.  Elektrochem.,  1911  ;  Xernst  Festschrift,  1912) 
in  which  the  rotational  energy  is  connected  with  the  frequencies 
of  the  infra-red  absorption  bands  of  a  gas.  The  chemical 
constants  of  gases  have  also  recently  been  calculated  from 
kinetic  principles  by  Sackur  (Xernst  Festschrift,  405,  1912 ; 
Ann.  Phi/s.,  40,  67,  84,  1913). 

At  very  low  temperatures  the  rotational  energy,  being  subject 
to  the  law  of  ergonic  distribution,  will  vanish,  and  hence  Cr  will 
approach  the  value  fR ;  in  the  case  of  hydrogen  the  molecular 
heat  has  been  experimentally  found  by  Eucken  (1-'  to  have  the 
value  2'98  in  liquid  air. 

Nernst  also  concludes  that  the  specific  heats  of  liquids  tend 
to  very  small  values  at  low  temperatures,  since  according 
to  Tammann  (§  88)  liquids  pass  into  amorphous  solids  at 
low  temperatures,  and  the  latter  are  subject  to  the  ergonic 
distribution. 

Speculations  on  the  application  of  the  theory  of  ergons  to 
chemical  processes  are  also  put  forward.'1"  From  a  know- 
ledge of  the  r-values  of  all  the  atoms  in  a  molecule,  the  potential 
energy  of  the  latter  may,  in  simple  cases,  be  calculated,  and  this 
is,  at  low  temperatures,  equal  to  the  heat  of  dissociation. 

226.  Debye's  Theory  of  the  Solid  State. 

A  solid  body,  the  molecules  of  which  are  monatomic,  and  all 
vibrating  with  a  constant  frequency,  v,  is  called  an  Einstein's 
solid,  since  it  formed  the  subject  of  Einstein's  application  of  the 
theory  of  ergonic  distribution  considered  in  §  §  222 — 24.  The 
equation  for  the  vibrational  energy 


-T-l 


so  derived  does  not  adequately  represent  the  experimental  data, 
and  was  replaced  by  the  Nernst-Lindemann  equation,  which,  how- 
ever, although  it  represents  the  change  of  specific  heat  with 
temperature  with  sufficient  accuracy,  possesses  the  disadvantage 
of  being  empirical. 

It  has  recently  been  shown  by  Debye  '•"'  that  an  equation  may 
be  obtained  which  not  only  represents  the  experimental  results 


530  THERMODYNAMICS 

better  than  the  Nernst-Lindemann  equation,  but  also  has  a 
theoretical  foundation.  He  supposes  that,  instead  of  a  single 
frequency,  the  body  is  characterised  by  a  complete  "  spectrum  " 
of  frequencies,  composed  of  a  finite  number  of  "  lines,"  this 
number  being  in  fact  equal  to  three  times  the  number  of  atoms 
in  the  body.  This  spectrum  is  characterised  by  the  density  of 
the  lines  per  interval  of  frequencies  dv  :  the  number  of  lines  fall- 
ing in  this  interval  is  found  to  be  proportional  to  v2,  where  the 
interval  extends  from  v  to  v-\-dv  : 

dz=3VVv*dv         .         .         .         .         (1) 

V  is  the  volume,  and  F  is  a  factor  of  proportionality,  which  is 
calculable  from  the  elastic  properties  of  the  solid.  The  connec- 
tion with  elasticity  was  in  fact  suspected  by  Sutherland  in  1910 
(Phil.  May.,  20,  657),  who  found  that  the  infra-red  frequency  of 
a  solid  was  of  the  same  order  as  the  frequency  of  an  elastic 
transversal  vibration  with  a  wave  length  equal  to  the  distance 
between  two  neighbouring  atoms.  To  every  degree  of  freedom 
Debve  assigns  an  amount  of  energy  : 


_  -i 


exactly  as  in  Einstein's  theory,  in  which  each  frequency  in  the 
series  obtains  an  amount  given  by  substituting  its  r-  value  in 
the  expression.  The  result  is  an  equation  for  the  energy, 
which  on  differentiation  with  respect  to  T  gives  the  specific  heat. 
The  latter  equation  is  written  by  Nernst  in  the  form  : 


C,.  =  3E 


• 


(2) 


At  low  temperatures  the  first  term  only  will  be  significant,  and 
hence  the  theory  leads  to  the  striking  result  that  at  very  low 
temperatures  the  specific  heat  of  a  solid  is  proportional  to  the  cube 
of  the  absolute  temperatiw,  or  its  energy  to  the  fourth  power  of  the 


KINETIC   THEORIES   IN   THERMODYNAMICS      587 

absolute  temperature.  The  latter  relation  holds  for  radiation  at 
all  temperatures  (Stefan-Boltzmanu  law).  Thus,  the  specific 
heats  vanish  less  rapidly  as  T  approaches  zero  than  they  would 
according  to  the  Nernst-Liudemann  equation,  which  requires  an 
exponentially  vanishing  specific  heat.  The  actually  observed 
relation  appears  to  lie  between  the  two.* 

The  theory  of  Debye  is  certainly  the  most  complete  and  suc- 
cessful attempt  to  represent  the  thermal  properties  of  solids 
which  has  yet  been  made  by  the  aid  of  the  theory  of  ergon ic 
distribution. 

227.     References  to  Chap.  XVIII. 

(1)  L.  Boltzmann,  Gastheorie,  ILL,  128. 

(2)  W.  Xernst,  Zeitschr.  Elektrochem.,  17,  270,  1911. 

(3)  L.   Boltzmann,    Wiener   Sitztingsber.,   03,   (2),    731 ;    cf.    F.   Eicharz, 
3farlnrg.  Sitzungsber.,  1907,  p.  93:    Zeitschr.  anorf/.  Chem.,  .;$,  356;  •>.'/,  14(i. 
1908 ;    A.  Wigand,  Ann.  Phys.,  [4],  22,  79,  1907. 

(4)  M.  Planck,  Vorlesungen  iiber  die  Theorie  der  Warme*trahlnn<j,  Leipzig. 
1906;   Acht  Vorlesungen  fiber  theoretische  Physik,  Leipzig,  1910;   Sitzungsber. 
KoniyL  Preuss.  ATcad.,  1911,  p.  723. 

(5)'  A.  Einstein,  Ann.  Phys.,  22,  180,  800,  1907 :  .;.},  170,  1911 :  cf.  J.  W. 
Gibbs,  Principles  of  Statistical  Mechanics  ;  F.  Hasenuhrl,  Phys.  Zeit&j,,-.,  J..', 
931,  1911 ;  P.  Frank,  ibid.,  13,  506,  1912  ;  F.  A.  Lindemann ,  Dissertation  : 
Ueber  das  Dulang-Petitsche  Oesetz,  Berlin,  1911. 

(6)  Lindemann    and  Magnus,  Zeit*chr.-Elektrocl,em.,    in,  269.    1910;  cf. 
Pollitzer,  ibid.,  17,  269,  1911. 

(7)  Lindemann  and  Nernst,  Berl.  Ber.,  1911,  p.  494 ;  Zeitschr.  ElektrocJxm., 
17,  817,  1911. 

(8)  Lindemann,  Phys.  Zeitschr.,  11,  609,  1910. 

(9)  A.  Bussell,  Phys.  Zeitschr.,  13,  59,  1912. 

(10)  F.  Koref,  Phys.  Zeitschr.  13,  183,  1912. 

(11)  Xernst,  Journ.  de  chim.  phys.,  8,  234,  1910;  Theoret.  Chem.,   1913. 
751;  cf.  O.  Sackur,  Ann.  Phys.,  [4],  .^,455,  1911  ;  Nernst,  Berl.  Ber.,  1911, 
Heft  4;  F.  Juttner,  Zeitschr.  Elektrochem.,  17,  139,  731,  1911. 

(12)  Eucken,  Berl.  Ber.,  1912,  p.  141 ;  cf.  Bjerrum.  Zritsciir.  Elektrochtm., 
17,  1911. 

(13)  Lindemann,  VerhL  Detitsch  Phys.  Ges.,  [13],  2$,  1107,  1911:    Haber 
and  Just,  Phys.  Zeitschr.,  12,  1035;  Ann.  Phys.,  [4],  368,  308,  1911. 

(14)  Debye,  Ann.  Phys.,  39,  789,  1912;  cf.  Xernst  and  Lindemann,  Ber?. 
Ber.  J-',  1912  ;  Born  and  Karman,  Phys.  Zeitschr.  1912. 

*  The  considerations  on  p.  530  require  revision  in  the  light  of  Debye's  theory. 


INDEX 


(The  numbers  refer  to  pages.) 


ABSOLUTE  zero,  64 

Absorption  coefficient,  276 

Acceleration,  21 

Accumulator,  468 

Adiabatic  change,  37,  75,  82,  126, 

127,  142,  148,  185 
Adsorption,     433,     446 ;     excess, 

435 ;    of  electrolytes,   473  ;    of 

gases  on  solids,  '440  ;    formula 

of  Gibbs,  436 
Adynamic  change,  37,  97 
J51o tropic  bodies,  193 
Affinity,  257,  506 
Allotropic  change,  19,  198,  487 
Amagat's  researches  on  gases,  154, 

226 

Amplitude  of  a  process,  114. 
Andrew's  diagram,  173 
A  niso tropic  bodies,  193 
Aphorism  of  Clausius,  83,  92 
Arrhenius'   theory  of  electrolytic 

dissociation,  301 
Aschistic  process,  31,  36,  51 
Atmosphere,  39 
Atomic  energy,  26 
Availability,  65,  66 
Available  energy,  66,  77.  80,  98, 

101 


BABO'S  law,  290,  395 

Battelli's  equation,  222 

Berthelot's  principle  (see  Thorn- 
sen)  ;  equation  for  gases.  156. 
236 

Bertrand's  vapour-pressure  equa- 
tion, 180 

Binary  mixtures.  381,  402,  403, 
410 

Biot's  vapour-pressure  equation, 
179 

Bodlander's  calculation,  509 

Boiling-point,  177,  289,  294 

Bolometer,  26 


Boltzmann's    theory    of    specific 

heats,  517 

Bound  energy,  98,  109 
Boyle's  law.  131  ;  deviations  from, 

152 
Brownian  movement,  70,  285 


CALORIC,  4,  69 

Calorie,  4,  30 

Calorimetry,  1 

Capacity,  for  heat,  5,  7.  108 ; 
factors,  111 

r'arnot's,  cyclic  process.  48.  54, 
71,  78,  149  ;  theorem.  58,  65, 
101 

Catalyst,  323,  355 

Characteristic  equation.  43  :  func- 
tions, 101 

Chemical  affinity.  257  ;  change, 
26,  75,  98,  253,  535  ;  constant, 
494, 496,  502  ;  energy.  26 :  equili- 
brium. 322 

Circuit,  44 

Clapeyron's  equation.  123  :  Clau- 
sius equation.  176,  244,  412, 
495 

Clark  cell,  510 

Classical  thermodynamics.  483 

Clausius's  characteristic  equation, 
222,  252 

Cold  reservoh-,  53 

Collision,  86 

Colloids.  70.  446,  447 

Compatibility  of  equilibria.  213, 
388 

Compensated  heat,  96 

Compensating  changes,  83 

Components,  169 

Compressibility,  40,  158,  486 

Compression,  influence  of,  199 

Concentration,  263 ;  cells,  463. 
465 

Condensed  systems,  484,  490 


540 


INDEX 


Conditions  of  equilibrium,  92 
Conduction  of  heat,  48,  84,  454 
Configuration,  22,  107 
Connodat  curve,  243 
Conservation  of  energy,  35 
Contact  potential  differences,  470 
Continuity  of  states,  174 
Corresponding  states,  228,  237 
Creighton.     See  Southern. 
Criteria  of  equilibrium,  93,  97.  99 
Critical  coefficient,  174  ;  constants 
173,  226,  236  ;  phenomena,  172, 
181,  188,  204,  246,  431  ;    solu- 
tion temperature,   407  ;    state, 
174  ;    temperature  of  mixtures, 
407 

Crystallisation,  194 
dimming  effect,  451 
Cyclic  process,  31,  36,  44,  73 


DALTON'S  law  of  partial  pressures. 

265,    277  ;      and    Gay-Lussac's 

law,  131,  160 
Debye's  theoiy  of  the  solid  state. 

535 

Degrees  of  freedom,  107,  169 
Demon,  70,  86,  88 
Density,    38,    44,    356;     critical, 

173  ;    of  saturated  vapour,  179 
Diagrams,  45,  128.  423 
Dieterici's     characteristic     equa- 
tions, 222,  236,  252  ;    freezing- 
point  equation,  420 
Difference  of  specific  heats,  125, 

138,  141,  526 

Differential  equations,  102 
Diffusion,  48,  75,  86,  268 
Dilution  law,  370 
Dirichlet's  theorem,  91 
Dissipation  of  energy,  66,  80,  84. 

99,  271 
Dissociation,    19,    207,    213,    301. 

328,    340,    506 ;     curves,    351  ; 

of  electrolytes,  301,  369 
Distillation,    fractional,    386  ;     of 

liquid  mixtures,  386,  415 
Distribution  law,  313,  367  ;   ratios 

of  ions,  473 
Dolezalek's    theory    of    mixtures, 

402  ;   rule,  403 
Double-layer,  454,  470 
Duhem's  theorem,  219  ;  Margule's 

equation,  395 
Diihring's  rule,  180 


Dulong's  rule,    142;     and  Petit's 

law,  15,  528 
Dyne,  22 


EFFICIENCY,  53 

Einstein's  theory  of  the  solid  state, 

521 

Elasticity,  40,  119,  486,  536 
Electrical  energy,  27 
Electrification,  influence  of,  204 
Electro-affinity,  475 
Electromolecular  force,  454 
Electromotive  force,  455,  476,  482, 

508  ;   and  chemical  equilibrium, 

477  ;  effect  of  pressure  on,  461  ; 

effect  of  temperature  on,  456 
Electron,  454,  514,  527 
Element,  388 
Endothermic  change,  256 
Energetics,  111 
Energiequantum,  112,  521 
Energy,  23,  25,  484,  513 
Entropy,  71,  75,  76,  79,  99,  101, 

484,  '531  ;     of  electricity,   454  ; 

electromagnetic,  520  ;   tempera- 
ture diagram,  76  ;    of  a  voltaic 

cell,  459 

Eotvos'  theorem,  431 
Equality  of  Clausius,  73 
Equilibrium,  20,  32,  50.  90,   122, 

169,  213,  322.  324,  353.  369,  374, 

447,  477,  497,  503 
Equipartition  of  energy,  519 
Equivalent  change,  83 
Erg,    23  ;     ergon,    521  ;     ergonic 

distribution,  524 
Euler's  theorem  on  homogeneous 

functions,  360 
Eutectic,  417 
Evaporation,    19,    171,    175,    213, 

231,  491 

Exothermic  reactions,  256 
Expansion,  41,  85,  485 
Explosions,  10,  147,  355 


FALSK  equilibrium,  90,  198 

Faraday's  laws,  455 

Findlay  s  rule,  307 

First  law  of  thermodynamics,  21, 

31,  73 

Flames.  355 

Fluids,  39,  117,  121,  124,  129 
Fluorescence,  26 


INDEX 


541 


Force,  21,  513;  function,  99; 
potential,  99 

Forcraud's  rule,  235 

Forms  of  energy,  25,  27 

Free  energy,  96,  101,  105,  109.. 
129,  470;  equation,  106;  sur- 
face, 424 

Friction,  25,  48,  75,  91 

Functions,  thermodynaiuic,  101 

Fundamental  differential  equa- 
tions, 102 

Fusion,  19,  192,  195,  213,  491 


GAS,  cells,  464,  477,  511  ;  charac- 
teristic equation,  131,  239 ; 
constant,  133,  134 ;  density, 
133  ;  entropy,  149 ;  equili- 
brium, 324,  353,  355,  497  ;  free 
energy,  151  ;  ideal,  135,  1395 
145  •  inert,  326  ;  kinetic  theory 
515;  mixtures,  263,  325  ;  mole- 
cular weight,  157  ;  potential, 
151  ;  temperature,  140  ;  velo- 
city of  sound  in,  146 

Generalised  co-ordinates,  107 

Gibbs's  adsorption  formula,  436  ; 
criteria  of  equilibrium  and 
stability,  93,  101  :  dissociation 
formula,  340,  499  ;  Helmholtz 
equation,  456,  460,  476  ;  Kono- 
walow  rule,  384,  416;  model, 
240 ;  paradox,  274 ;  phase 
rule,  169,  388  ;  theorem,  220. 

Graetz'  vapour-pressure  equation, 
191 

Graphical  representation,  423 

Gravity,  21,  201 

Griineisen's  relation,  486 

Guldberg's  rale,  234 


HEAT  of  admixture,  394,  406; 
of  adsorption,  444;  capacity, 
5,  7,  108  ;  of  dilution,  312,  391  ; 
of  dissociation,  340,  373  ;  energy 
25  ;  engines,  53  ;  of  evapora- 
tion (internal),  183,  433 ;  of 
formation,  255 ;  function,  42, 
101  ;  of  ionisation,  477  ;  latent, 
18.  98,  118,  123,  299,  430,  457  ; 
of  reaction,  255,  259,  507; 
of  solution,  302,  310,  373,  395  ; 
of  swelling,  447  ;  unit,  4,  30  ; 
of  volatilisation,  392 


Heat,  specific,  5,  7,  117,  485 ; 
of  electricity,  451  ;  of  gases,  9, 
142,  349  (dissociating),  515, 
535;  of  liquids,  17,  535;  of 
solids,  12,  16,  485,  517,  521, 
526 ;  of  solutions,  17  ;  of 
vapours,  12 

Henry's  law,  275,  372 

Hertz's  vapour-pressure  equation, 
191 

Hess's  principle,  254 

Heterogeneous  equilibria,  169, 375, 
503 

Hot  reservoir,  53 


ICE  type,  195 
Ideal  gas,  47,  135 
Independent  variables,  103 
Indicator  diagram,  45,  127 
Inequality  of  Clausius,  79 
Intensity 'factors,  111 
Intrinsic  energy,  32,  76,  484 
Inversion  point,  167 
Irreversible  processes,  67,  69,  75, 

82,  84,  87 

Isentropic  change.  75 
Isochore,  44,  337 
Isolated  system,  37 
Isomorphous  mixture,  417 
Isopiestic  change,  44,  337 
Isopneuma,  442 
Isosteres,  442 
Isotherm,    44,     127  ;     isothermal 

change,     95.     125,     142,     148; 

cycles,  60 
Isotropic  bodies,  193 


JOCHM ANN'S  equation,  164 
Joule,  31  ;  experiments  with,  gases, 

137  ;    Kelvin  effect,   164,  225  ; 

researches,  28,  51 ;  theorem,  136 

KIN-ETIC,  energy,  24;  theory  of 
dissipation,  87  ;  theory  of  gases, 
515 ;  theory  of  solids,  517  ; 
theories  in  thermodynamics,  513 

Kirchoff s  equation  for  effect  of 
temperature,  112.  259;  equa- 
tions for  vapour-pressure,  179, 
190,  192,  390,  412 

Konowalow's  theorem,  385,  407  ; 
vapour-pressure  curves,  382 


542 


INDEX 


LAPLACE'S  equation,  146 
Least  action,  principle  of,  69,  304 
Line  of  heterogeneous  states,  172 
Liquefaction  of  gases,   167,    173  ; 

of  mixtures,  428 
Liquids,  miscible,  382  ;    partially 

miscible,  406  ;   immiscible,  409 
Litre-atmosphere,  48 
Luther's  rule,  480 


MACHINES,  25,  52 

Macro-differential,  39 

Magnetic  energy,  27 

Magnus's  vapour-pressure  equa- 
tion, 180 

Mass,  22  ;   action,  328,  329,  367 

Massieu's  theorems,  129 

Maximum  work,  65,  77,  102.  112, 
470,  489,  507  ;  of  a  gas  reaction, 
330  ;  influence  of  temperature 
on,  348  ;  principle  of,  258 

Maxwell's  principle,  128  ;  rela- 
tions, 104  ;  rule,  182,  243 

Mayer's  calculation,  28,  136 

Mean  values,  39 

Mechanical  equivalent  of  heat,  28, 
30,  136 

Mechanics,  21 

Melting  point,  193,  203,  528 

Meslin's  theorem,  229 

Metastable  states,  181 

Mixed  liquids,  380 

Mixture  rule,  263 

Mobile  equilibrium,  304,  340 

Model,  thermodynamic,  240 

Mol,  20,  135 

Molecular  complexity,  143  ;  hypo- 
theses, 513  ;  physics,  38  ;  vor- 
tices, 69  ;  weight,  133 

Motion,  22 

Motivation,  92 

Motivity,  78,  85,  101 

Moutier's  theorem,  60,  73,  212, 
217 


NEBULA,  68 

Nernst  equation  for  concentration 

cells,  467  ;    theorem,  484,  489, 

508,   531  ;     theory   of  galvanic 

cells,  474 
Nernst-Lindemann    equation    for 

specific  heats,  527 
Neumann-Kopp  rule,  16,  489 


Normal    substances,    238 ;     vari- 
ables, 107 


OSMOTIC  pressure,  279,  288 
Ostwald's  dilution  law,  370 
Oxidation  and  reduction,  479 


PAPIN'S  digester,  177 

Partial  pressures,  law  of,  171, 
265,  274 

Pascal's  law,  40 

Passive  resistances,  91 

Path,  44 

Pawlewski's  rule,  407 

Peltier  effect,  450,  460 

Perpetuum  mobile,  51,  70 

Phase,  20  ;   rule,  169,  388,  446 

Phosphorescence,  35 

Physically  small,  38,  69 

Plait  point,  244 

Planck's  equation,  237  ;  for 
saturated  vapour,  189  ;  poten- 
tial, 102  ;  theory  of  radiation, 
520 

Polarisation,  481 

Polymorphic  change,  198,  213 

Porous-plug  experiment,  138,  162 

Potential,  chemical,  329,  358,  361  ; 
diagram,  424 ;  energy,  23 ; 
equation,  106,  110,  129,  326; 
thermodvnamic,  95,  99,  105, 
129 

Pressure,  37,  39 

Probability,  87 


RADIANT  energy,  25,  26 

Radiation,  48,  85,  89,  520,  524 

Radioactive  changes,  35,  68  ; 
energy,  26 

Ramsay  and  Young's  equation, 
180 

Range  of  a  process,  114 

Rankine's  cycle,  113  ;  vapour- 
pressure  equation,  179 

Raoult's  freezing-point  law,  299  ; 
vapour-pressure  law,  291 

Ratio  of  specific  heats,  118,  143, 
144,  516 

Rayleigh's  separation  of  gas  mix- 
tures, 272  ;  radiation  equation, 
524 


INDEX 


543 


Reaction  energy,  98  ;    Le  Chate- 

lier's  principle  of,  304 
Reciprocal  relations,  104 
Reduced  magnitudes,  229 
Reech's  theorem,  118,  144 
Refrigerator,  53 
Restoration  of  energy,  68 
Reversibility,  48 
Reversible  engine,  54,  58,  59,  72  ; 

process,  48,  82  ;    reactions,  323 
Robin's  theorem,  211 
Roozeboom's  theorem,  217,  220 
Rotational  energy,  534 


SALT-HYDRATES,  379,  427 

Sarrau's  principle,  251 

Saturated  vapour,  density  of,  179 

Saturation  curve,  210 

Schistic  process,  32 

Second  law  of  thermodvnamics,  39, 

51,52,68,  73,86,  112 
Semipermeable    septa,    272,    279, 

287,  356 

Simultaneous  equilibria.  213 
Single  potential- differences,   474 
Softening,  193 

Solubility,  causes  modifying,  319 
and    chemical    potential,    359 
curve,     307  ;      equation,     306 
of  gases  in  liquids,   275,   371 
effect  of  pressure  on,  316  ;  effect 
of    surface    tension    on,    447 ; 
effect  of  temperature  on,  302, 
372 

Solutions,  dilute,  boiling  points  of, 
289,  294  ;  equilibrium  in,  363, 
366,  367  ;  freezing-points  of, 
296,  374  ;  osmotic  pressure  of, 
282  ;  potential  of,  366  ;  solid, 
320,  502  ;  vapour  pressure  of, 

288,  374 ;    of  gases  in  liquids, 
275 ;     freezing-points    of,    418, 
427 ;     solid,    417 ;     theory    of, 
262,  285,  410,  502 

Supertension,  481 
Sorting-demon,  70 
Source,  53 

Southern  and  Creigh ton's  laws,  184 
Spinodal  curve,  245 
Spinthariscope,  38 
Stability,  90 

Stafen-Boltzinann    law    of   radia- 
tion, 536 
State,  32,  43,  88,  123,  207 


Stationary  motions,  69 

Statistical  method,  69 

Streaming  method,  354 

Stress,  21,  38 

Sublimation,  19,  191,  491 

Substance,  43 

Surface,  energy,  27 ;  tension,  202, 

429,  447  ;  thermodynamic,  101, 

128,  240 

Suspended  transition,  177 
Systems,  heterogeneous,  170 


TAMMANN'S  researches  on  fusion, 
205 

Temperature,  1,  4,  60,  140,  162 

Thermal,  co-efficients,  117,  223; 
relations  of  binary  mixtures, 
403 

Thermochemistry,  254,  507 

Thermodynamic*  potentials,  99 

Thermo-electric  circuit,  450  ;  in- 
version, 451  ;  theories,  453 

Thermometers,  3,  140,  166 

Therinometry,  1,  353 

Thomsen-Berthelot  principle,  257, 
506 

Thomson  rule,  459,  510  ;  isotherm, 
182,  227,  243 

Transition,  19,  309,  461,  487 

Trevelyau  rocker,  25 

Triple  point,  214,  252 

Trouton's  rule,  234 

True  equilibrium,  90 


UNAVAILABLE  energy,  77 
Uncompensated  heat,  96 
Universe,  68,  83 


VAN  DER  WAAL>  characteristic 
equation,  221  ;  vapour- pressure 
equation,  180 

Van't  Hoff's  boiling-point  equa- 
tion, 295  ;  freezing-point  equa- 
tion, 299  ;  theory  of  solution, 
287  ;  principle  of  mobile  equili- 
brium, 304,  340 

Variance,  169 

Vapour,  energy  and  entropy  of, 
183;  pressure,  171,  180,  201  ; 
curves,  382,  399,  408;  of 
dilute  solutions,  288  ;  equations, 
179,  190,  492 


544 


INDEX 


Velocity  of  sound,  146 
Virtual  change,  92  ;  work,  50 
Viscosity,  87 
Vital  processes,  35,  70 
Voltaic  cell,  53,  357,  455 


WATT'S  rule,  184 


Wax -type  of  fusion,  195 

Weston  normal  element,  456 

Welding,  194 

Wetting,  445 

Woestyn's  law,  16 

Work,    21,    22,    41,    45,    47,    147, 

349 
Working  substance,  53 


.,  PRINTERS,  LONDON  ANP  TOKB1UDGE. 


VAN  NOSTRAND'S 
"WESTMINSTER"  SERIES 

Bound  in  Uniform  Style. 
Fully  Illustrated.        Price  S2.OO  net  each. 

Gas  Engines.  By  W.  J.  MARSHALL,  Assoc.  M.I.Mech.E., 
and  CAPT.  H.  RIALL  SANKEY,  R.E.  (Ret.).  M.Inst.C.E., 
M.I.Mech.E.  300  Pages,  127  Illustrations. 
LIST  OF  CONTENTS  :  Theory  of  the  Gas  Engine.  The  Otto  Cycle.  The 
Two  Stroke  Cycle.  Water  Cooling  of  Gas  Engine  Parts.  Ignition. 
Operating  Gas  Engines.  The  Arrangement  of  a  Gas  Engine  Instal- 
lation. The  Testing  of  Gas  Engines.  Governing.  Gas  and  Gas 
Producers.  Index. 

Textiles*  By  A.  F.  BARKER,  M.Sc.,  with  Chapters  on  the 
Mercerized  and  Artificial  Fibres,  and  the  Dyeing  of 
Textile  Materials  by  W.  M.  GARDNER,  M.Sc.,  F.C.S. ; 
Silk  Throwing  and  Spinning,  by  R.  SNOW;  the  Cotton 
Industry,  by  W.  H.  COOK  ;  the  Linen  Industry,  by  F. 
BRADBURY.  370  Pages.  86  Illustrations. 

CONTENTS  :  The  History  of  the  Textile  Industries  ;  also  of  Textile 
Inventions  and  Inventors.  The  Wool,  Silk,  Cotton,  Flax,  etc.. 
Growing  Industries.  The  Mercerized  and  Artificial  Fibres  em- 
ployed in  the  Textile  Industries.  The  Dyeing  of  Textile  Materials. 
The  Principles  of  Spinning.  Processes  preparatory  to  Spinning. 
The  Principles  of  Weaving.  The  Principles  of  Designing  and 
Colouring.  The  Principles  of  Finishing.  Textile  Calculations. 
The  Woollen  Industry.  The  Worsted  Industry.  The  Dress 
Goods,  Stuff,  and  Linings  Industry.  The  Tapestry  and  Carpet 
Industry.  Silk  Throwing  and  Spinning,  The  Cotton  Industry. 
The  Linen  Industry  historically  and  commercially  considered. 
Recent  Developments  and  the  Future  of  the  Textile  Industries. 
Index. 

Soils  and  Manures.    By  J.  ALAN  MURRAY,  B.Sc.    367 

Pages.     33  Illustrations. 

CONTENTS  :  Introductory.  The  Origin  of  Soils.  Physical  Proper- 
ties of  Soils.  Chemistry  of  Soils.  Biology  of  Soils.  Fertility. 
Principles  of  Manuring.  Phosphatic  Manures.  Phosphonitro- 
genous  Manures.  Nitrogenous  Manures.  Potash  Manures. 
Compound  and  Miscellaneous  Manures.  General  Manures.  Farm- 
yard Manure.  Valuation  of  Manures.  Composition  and  ManuraJ 
Value  of  Various  Farm  Foods. 


THE    "WESTMINSTER"    SERIES 


Coal.  By  JAMES  TONGE,  M.I.M.E.,  F.G.S.,  etc.  (Lecturer 
on  Mining  at  Victoria  University,  Manchester).  283 
Pages.  With  46  Illustrations,  many  of  them  showing  the 
Fossils  found  in  the  Coal  Measures. 

LIST  OF  CONTENTS  :  History.  Occurrence.  Mode  of  Formation 
of  Coal  Seams.  Fossils  of  the  Coal  Measures.  Botany  of  the 
Coal-Measure  Plants.  Coalfields  of  the  British  Isles.  Foreign 
Coalfields.  The  Classification  of  Coals.  The  Valuation  of  Coal. 
Foreign  Coals  and  their  Values.  Uses  of  Coil.  The  Production 
of  Heat  from  Coal.  Waste  of  Coal.  The  Preparation  of  Coal 
for  the  Market.  Coaling  Stations  of  the  World.  Index. 

Iron  and  Steel    By  J.  H.  STANSBIE,  B.Sc.  (Lond.),  F.I.C. 

385  Pages.     With  86  Illustrations. 

LIST  OF  CONTENTS  :  Introductory.  Iron  Ores.  Combustible  and 
other  materials  used  in  Iron  and  Steel  Manufacture.  Primitive 
Methods  of  Iron  and  Steel  Production.  Pig  Iron  and  its  Manu- 
facture. The  Refining  of  Pig  Iron  in  Small  Charges.  Crucible 
and  Weld  Steel.  The  Bessemer  Process.  The  Open  Hearth 
Process.  Mechanical  Treatment  of  Iron  and  Steel.  Physical 
and  Mechanical  Properties  of  Iron  and  Steel.  Iron  and  Steel 
under  the  Microscope.  Heat  Treatment  of  Iron  and  Steel.  Elec- 
tric Smelting.  Special  Steels.  Index. 

Timber,     By  J.  R.    BATERDEN,  Assoc.M.Inst.C.E.     334 

Pages.     54  Illustrations. 

CONTENTS  :  Timber.  The  World's  Forest  Supply.  Quantities  of 
Timber  used.  Timber  imports  into  Great  Britain.  European 
Timber.  Timber  of  the  United  States  and  Canada.  Timbers 
of  South  America,  Central  America,  and  West  India  Islands.  Tim- 
bers of  India,  Burma,  and  Andaman  Islands.  Timber  of  the 
Straits  Settlements,  Malay  Peninsula,  Japan  and  South  and 
West  Africa.  Australian  Timbers.  Timbers  of  New  Zealand 
and  Tasmania.  Causes  of  Decay  and  Destruction  of  Timber. 
Seasoning  and  Impregnation  of  Timber.  Defects  in  Timber  and 
General  Notes.  Strength  and  Testing  of  Timber.  "  Figure  "  in 
Timber.  Appendix.  Bibliography. 

Natural  Sources  of  Power.  By  ROBERT  S.  BALL,  B.Sc., 
A.M.Inst.C.E.  362  Pages.  With  104  Diagrams  and 
Illustrations. 

CONTENTS  :  Preface.  Units  with  Metric  Equivalents  and  Abbre- 
viations. Length  and  Distance.  Surface  and  Area.  Volumes. 
Weights  or  Measures.  Pressures.  Linear  Velocities,  Angular 
Velocities.  Acceleration.  Energy.  Power.  Introductory 
Water  Power  and  Methods  of  Measuring.  Application  of  Water 
Power  to  the  Propulsion  of  Machinery.  The  Hydraulic  Turbine, 
(  2  ) 


THE    "WESTMINSTER"    SERIES 


Various  Types  of  Turbine.  Construction  of  Water  Power  Plants. 
Water  Power  Installations.  The  Regulation  of  Turbines.  Wind 
Pressure.  Velocity,  and  Methods  of  Measuring.  The  Application 
of  Wind  Power  to  Industry.  The  Modern  Windmill.  Con- 
structional Details.  Power  of  Modern  Windmills.  Appendices. 
A.B.C  Index. 

Electric   Lamps.      By    MAURICE    SOLOMON,    A.C.G.T., 

A.M.I.E.E.  339  Pages.  112  Illustrations. 
CONTENTS  :  The  Principles  of  Artificial  Illumination.  The  Produc- 
tion of  Artificial  Illumination.  Photometry.  Methods  of  Testing. 
Carbon  Filament  Lamps.  The  Nernst  Lamp.  Metallic  Filament 
Lamps.  The  Electric  Arc.  The  Manufacture  and  Testing  of  Arc 
Lamp  Carbons.  Arc  Lamps.  Miscellaneous  Lamps.  Compari- 
son of  Lamps  of  Different  Types. 

Liquid  and  Gaseous  Fuels,  and  the  Part  they  play 
in  Modern  Power  Production.  By  Professor 
VIVIAN  B.  LEWES,  F.I.C.,  F.C.S.,  Prof,  of  Chemistry, 
Royal  Naval  College,  Greenwich.  350  Pages.  With  54 
Illustrations. 

LIST  OF  CONTENTS  :  Lavoisier's  Discovery  of  the  Nature  of  Com- 
bustion, etc.  The  Cycle  of  Animal  and  Vegetable  Life.  Method 
of  determining  Calorific  Value.  The  Discovery  of  Petroleum 
in  America.  Oil  Lamps,  etc.  The  History  of  Coal  Gas.  Calorific 
Value  of  Coal  Gas  and  its  Constituents.  The  History  of  Water 
Gas.  Incomplete  Combustion.  Comparison  of  the  Thermal 
Values  of  our  Fuels,  etc.  Appendix.  Bibliography.  Index. 

Electric   Power   and    Traction.     By  F.  H.  DAVIES, 

A.M.T.E.E.  299  Pages.  With  66  Illustrations. 
LIST  OF  CONTENTS  :  Introduction.  The  Generation  and  Distri- 
bution of  Power.  The  Electric  Motor.  The  Application  of 
Electric  Power.  Electric  Power  in  Collieries.  Electric  Power 
in  Engineering  Workshops.  Electric  Power  in  Textile  Factories. 
Electric  Power  in  the  Printing  Trade.  Electric  Power  at  Sea. 
Electric  Power  on  Canals.  Electric  Traction.  The  Overhead 
System  and  Track  Work.  The  Conduit  System.  The  Surface 
Contact  System.  Car  Building  and  Equipment.  Electric  Rail- 
ways. Glossary.  Index. 

Decorative    Glass    Processes.      By    ARTHUR    Louis 

DUTHIE.     279  Pages.     38  Illustrations. 

CONTENTS  :  Introduction.  Viuious  Kinds  of  Glass  in  Use  :  Their 
Characteristics,  Comparative  Price,  etc.  Leaded  Lights.  Stained 
Glass.  Embossed  Glass.  Brilliant  Cutting  and  Bevelling.  Sand- 
Blast  and  Crystalline  Giass.  Gilding.  Silvering  and  Mosa/c. 
Proprietary  Processes.  Patents.  Glossary. 
(  3  ) 


THE    "WESTMINSTER'*    SERIES 


Town  Gas  and  its  Uses  for  the  Production  of 
Light,  Heat,  and  Motive  Power.  By  W.  H.  Y. 
WEBBER,  C.E.  282  Pages.  With  71  Illustrations. 
LIST  OF  CONTENTS  :  The  Nature  and  Properties  of  Town  Gas.  The 
History  and  Manufacture  of  Town  Gas.  The  Bye-Products  of 
Coal  Gas  Manufacture.  Gas  Lights  and  Lighting.  Practical 
Gas  Lighting.  The  Cost  of  Gas  Lighting.  Heating  and  Warm- 
ing by  Gas.  Cooking  by  Gas.  The  Healthfulness  and  Safety 
of  Gas  in  all  its  uses.  Town  Gas  for  Power  Generation,  including 
Private  Electricity  Supply.  The  Legal  Relations  of  Gas  Sup- 
pliers, Consumers,  and  the  Public.  Index. 

Electro-Metallurgy.      By    J.    B.    C.    KERSHAW,   F.I.C. 

318  Pages.     With  61  Illustrations. 

CONTENTS  :  Introduction  and  Historical  Survey.  Aluminium. 
Production.  Details  of  Processes  and  Works.  Costs.  Utiliza- 
tion. Future  of  the  Metal.  Bullion  and  Gold.  Silver  Refining 
Process.  Gold  Refining  Processes.  Gold  Extraction  Processes. 
Calcium  Carbide  and  Acetylene  Gas.  The  Carbide  Furnace  and 
Process.  Production.  Utilization.  Carborundum.  Details  of 
Manufacture.  Properties  and  Uses.  Copper.  Copper  Refin- 
ing. Descriptions  of  Refineries.  Costs.  Properties  and  Utiliza- 
tion. The  Elmore  and  similar  Processes.  Electrolytic  Extrac- 
tion Processes.  Electro-Metallurgical  Concentration  Processes. 
Ferro-alloys.  Descriptions  of  Works.  Utilization.  Glass  and 
Quartz  Glass.  Graphite.  Details  of  Process.  Utilization.  Iron 
and  Steel.  Descriptions  of  Furnaces  and  Processes.  Yields  and 
Costs.  Comparative  Costs.  Lead.  The  Salom  Process.  The  Betts 
Refining  Process.  The  Betts  Reduction  Process.  White  Lead  Pro- 
cesses. Miscellaneous  Products.  Calcium.  Carbon  Bisulphide. 
Carbon  Tetra-Chloride.  Diamantine.  Magnesium.  Phosphorus. 
Silicon  and  its  Compounds.  Nickel.  Wet  Processes.  Dry 
Processes.  Sodium.  Descriptions  of  Cells  and  Processes.  Tin. 
Alkaline  Processes  for  Tin  Stripping.  Acid  Processes  for  Tin 
Stripping.  Salt  Processes  for  Tin  Stripping.  Zinc.  Wet  Pro- 
cesses. Dry  Processes.  Electro-Thermal  Processes.  Electro 
Galvanizing.  Glossary.  Name  Index. 

Radio-Telegraphy.    By  C.  C.   F.  MONCKTON,   M.I.E.E. 

389  Pages.  With  173  Diagrams  and  Illustrations. 
CONTENTS  :  Preface.  Electric  Phenomena.  Electric  Vibrations. 
Electro-Magnetic  Waves.  Modified  Hertz  Waves  used  in  Radio- 
Telegraphy.  Apparatus  used  for  Charging  the  Oscillator.  The 
Electric  Oscillator  :  Methods  of  Arrangement,  Practical  Details. 
The  Receiver :  Methods  of  Arrangement,  The  Detecting  Ap- 
paratus, and  other  details.  Measurements  in  Radio-Telegraphy. 
The  Experimental  Station  at  Elmers  End  :  Lodge-Muirhead 
System.  Radio  -  Telegraph  Station  at  Nauen  :  Telefunken 
System.  Station  at  Lyngby  :  Poulsen  System.  The  Lodge- 

(   4   ) 


THE    "WESTMINSTER"    SERIES 

Muirhead  System,  the  Marconi  System,  Telefunken  System,  and 
Poulsen  System.  Portable  Stations.  Radio-Telephony.  Ap- 
pendices :  The  Morse  Alphabet.  Electrical  Units  used  in  this 
Book.  International  Control  of  Radio-Telegraphy.  Index. 

India-Rubber  and  its  Manufacture,  with  Chapters 
on  Gutta-Percha  and  Balata.  By  H.  L.  TERRY, 
F.I.C.,  Assoc.Inst.M.M.  303  Pages.  With  Illustrations. 

LIST  OF  CONTENTS  :  Preface.  Introduction :  Historical  and 
General.  Raw  Rubber.  Botanical  Origin.  Tapping  the  Trees. 
Coagulation.  Principal  Raw  Rubbers  of  Commerce.  Pseudo- 
Rubbers.  Congo  Rubber.  General  Considerations.  Chemical 
and  Physical  Properties.  Vulcanization.  India-rubber  Planta- 
tions. India-rubber  Substitutes.  Reclaimed  Rubber.  Washing 
and  Drying  of  Raw  Rubber.  Compounding  of  Rubber.  Rubber 
Solvents  and  their  Recovery.  Rubber  Solution.  Fine  Cut  Sheet 
and  Articles  made  therefrom.  Elastic  Thread.  Mechanical 
Rubber  Goods.  Sundry  Rubber  Articles.  India-rubber  Proofed 
Textures.  Tyres.  India-rubber  Boots  and  Shoes.  Rubber  for 
Insulated  Wires.  Vulcanite  Contracts  for  India-rubber  Goods. 
The  Testing  of  Rubber  Goods.  Gutta-Percha.  Balata.  Biblio- 
graphy. Index. 

The  Railway  Locomotive.  What  It  Is,  and  Why  It  is 
What  It  Is.  By  VAUGHAN  PENDRED,  M.InstM.E., 
Mem.Inst.M.I.  321  Pages.  94  Illustrations. 

CONTENTS  :  The  Locomotive  Engine  as  a  Vehicle — Frames.  Bogies. 
The  Action  of  the  Bogie.  Centre  of  Gravity.  Wheels.  Wheel 
and  Rail.  Adhesion.  Propulsion.  Counter-Balancing.  The  Loco- 
motive as  a  Steam  Generator — The  Boiler.  The  Construction  of  the 
Boiler.  Stay  Bolts.  The  Fire-Box.  The  Design  of  Boilers. 
Combustion.  Fuel.  The  Front  End.  The  Blast  Pipe.  Steam 
Water.  Priming.  The  Quality  of  Steam.  Superheating.  Boiler 
Fittings.  The  Injector.  The  Locomotive  as  a  Steam  Engine — 
Cylinders  and  Valves.  Friction.  Valve  Gear.  Expansion.  The 
Stephenson  Link  Motion.  Walschaert's  and  Joy's  Gears.  Slide 
Valves.  Compounding.  Piston  Valves.  The  Indicator.  Ten- 
ders. Tank  Engines.  Lubrication.  Brakes.  The  Running  Shed. 
The  Work  of  the  Locomotive. 

Glass  Manufacture.  By  WALTER  ROSENHAIN,  Superin- 
tendent of  the  Department  of  Metallurgy  in  the  National 
Physical  Laboratory,  late  Scientific  Adviser  in  the  Glass 
Works  of  Messrs.  Chance  Bros.  &  Co.  280  Pages.  With 
Illustrations. 

CONTENTS  :  Preface.  Definitions.  Physical  and  Chemical  Qualities, 
Mechanical,  Thermal,  and  Electrical  Properties.  Transparency 

(   5    ) 


THE    "WESTMINSTER"    SERIES 


and  Colour.  Raw  materials  of  manufacture.  Crucibles  and 
Furnaces  for  Fusion.  Process  of  Fusion.  Processes  used  in 
Working  of  Glass.  Bottle.  Blown  and  Pressed.  Rolled  or 
Plate.  Sheet  and  Crown.  Coloured.  Optical  Glass :  Nature 
and  Properties,  Manufacture.  Miscellaneous  Products.  Ap- 
pendix. Bibliography  of  Glass  Manufacture.  Index 

Precious  Stones.    By  W.  GOODCHILD,  M.B.,  B.Ch.    319 
Pages.    With  42  lUustrations.    With  a  Chapter  on 
Artificial  Stones.    By  ROBERT  DYKES. 

LIST  OF  CONTENTS  :  Introductory  and  Historical.  Genesis  rf 
Precious  Stones.  Physical  Properties.  The  Cutting  and  Polish- 
ing of  Gems.  Imitation  Gems  and  the  Artificial  Production  of 
Precious  Stones.  The  Diamond.  Fluor  Spar  and  the  Forms  of 
Silica.  Corundum,  including  Ruby  and  Sapphire.  Spinel  and 
Chrysoberyl.  The  Carbonates  and  the  Felspars.  The  Pyroxene 
and  Amphibole  Groups.  Beryl,  Cordierite,  Lapis  Lazuli  and  the 
Garnets.  Olivine,  Topaz,  Tourmaline  and  other  Silicates.  Phos- 
phates, Sulphates,  and  Carbon  Compounds. 

INTRODUCTION  TO  THE 

Chemistry  and  Physics  of  Building  Materials. 
By  ALAN  E.  MUNBY,  M.A.  365  Pages.  Illustrated. 

CONTENTS  :  Elementary  Science  :  Natural  Laws  and  Scientific  In- 
vestigations. Measurement  and  the  Properties  of  Matter.  Air 
and  Combustion.  Nature  and  Measurement  of  Heat  and  Its 
Effects  on  Materials.  Chemical  Signs  and  Calculations.  Water 
and  Its  Impurities.  Sulphur  and  the  Nature  of  Acids  and  Bases. 
Coal  and  Its  Products.  Outlines  of  Geology.  Building  Materials  : 
The  Constituents  of  Stones,  Clays  and  Cementing  Materials.  Clas- 
sification, Examination  and  Testing  of  Stones,  Brick  and  Other 
Clays.  Kiln  Reactions  and  the  Properties  of  Burnt  Clays.  Plasters 
and  Limes.  Cements.  Theories  upon  the  Setting  of  Plasters  and 
Hydraulic  Materials.  Artificial  Stone.  Oxychloride  Cement. 
Asphaite.  General  Properties  of  Metals.  Iron  and  Steel.  Other 
Metals  and  Alloys.  Timber.  Paints :  Oils,  Thinners  and  Varnishes ; 
Bases,  Pigments  and  Driers. 

Patents,  Designs  and  Trade  Marks  :  The  Law 
and  Commercial  Usage.  By  KENNETH  R.  SWAN, 
B.A.  (Oxon.),  of  the  Inner  Temple,  Barrister-at-Law. 
402  Pages. 

CONTENTS  :    Table  of  Cases  Cited— Part  I. — Letters  Patent.     Intro- 
duction. General.     Historical.    I.,    II.,    III.   Invention,    Novelty, 
(   6    ) 


THE    "WESTMINSTER"    SERIES 

Subject  Matter,  and  Utility  the  Essentials  of  Patentable  Invention. 
IV.  Specification.  V.  Construction  of  Specification.  VI.  Who 
May  Apply  for  a  Patent.  VII.  Application  and  Grant.  VIII. 
Opposition.  IX.  Patent  Rights.  Legal  Value.  Commercial 
Value.  X.  Amendment.  XI.  Infringement  of  Patent.  XII. 
Action  for  Infringement.  XIII.  Action  to  Restrain  Threats. 
XIV.  Negotiation  of  Patents  by  Sale  and  Licence.  XV.  Limita- 
tions on  Patent  Right.  XVI.  Revocation.  XVII.  Prolonga- 
tion. XVIII.  Miscellaneous.  XIX.  Foreign  Patents.  XX. 
Foreign  Patent  Laws  :  United  States  of  America.  Germany. 
France.  Table  of  Cost,  etc.,  of  Foreign  Patents.  APPENDIX  A. — 
i.  Table  of  Forms  and  Fees.  2.  Cost  of  Obtaining  a  British 
Patent.  3.  Convention  Countries.  Part  II. — Copyright  in 
Design.  Introduction.  I.  Registrable  Designs.  II.  Registra- 
tion. III.  Marking.  IV.  Infringement.  APPENDIX  B. — I. 
Table  of  Forms  and  Fees.  2.  Classification  of  Goods.  Part 
III. — Trade  Marks.  Introduction.  I.  Meaning  of  Trade  Mark. 
II.  Qualification  for  Registration.  III.  Restrictions  on  Regis- 
tration. IV.  Registration.  V.  Effect  of  Registration.  VI. 
Miscellaneous.  APPENDIX  C. — Table  of  Forms  and  Fees.  INDICES. 
I.  Patents.  2.  Designs.  3.  Trade  Marks. 


The  Book:  Its  History  and  Development.  By 
CYRIL  DAVENPORT,  V.D.,  F.S.A.  266  Pages.  With 
7  Plates  and  126  Figures  in  the  text. 

LIST  OF  CONTENTS  :  Early  Records.  Rolls,  Books  and  Book 
bindings.  Paper.  Printing.  Illustrations.  Miscellanea. 
Leathers.  The  Ornamentation  of  Leather  Bookbindings  without 
Gold.  The  Ornamentation  of  Leather  Bookbindings  with  Gold. 
Bibliography.  Index. 

The  Manufacture  of  Paper.  By  R.  W.  SINDALL,  F.C.S., 
Consulting  Chemist  to  the  Wood  Pulp  and  Paper  Trades  ; 
Lecturer  on  Paper-making  for  the  Hertfordshire  County 
Council,  the  Bucks  County  Council,  the  Printing  and 
Stationery  Trades  at  Exeter  Hall  (1903-4),  the  Institute 
of  Printers ;  Technical  Adviser  to  the  Government  of 
India,  1905.  275  Pages.  58  Illustrations. 
CONTENTS  :  Preface.  List  of  Illustrations.  Historical  Notice.  Cel- 
lulose and  Paper-making  Fibres.  The  Manufacture  of  Paper  from 
Rags,  Esparto  and  Straw.  Wood  Pulp  and  Wood  Pulp  Papers. 
Brown  Papers  and  Boards.  Special  kinds  of  Paper.  Chemicals 
used  in  Paper-making.  The  Process  of  "  Beating.  The  Dye- 
ing and  Colouring  of  Paper  Pulp.  Paper  Mill  Machinery.  The 
Deterioration  of  Paper.  Bibliography.  Index. 
(  7  ) 


THE   "WESTMINSTER"   SERIES 


Wood  Pulp  and  its  Applications.  By  C.  F.  CROSS, 
B.Sc.,  F.I.C.,  E.  J.  BEVAN,  F.I.C.,  and  R.  W.  SINDALL, 
F.C.S.  266  pages.  36  Illustrations. 

CONTENTS:  The  Structural  Elements  of  Wood.  Cellulose  as  a 
Chemical.  Sources  of  Supply.  Mechanical  Wood  Pulp.  Chemical 
Wood  Pulp.  The  Bleaching  of  Wood  Pulp.  News  and  Printings. 
Wood  Pulp  Boards.  Utilisation  of  Wood  Waste.  Testing  of 
Wood  Pulp  for  Moisture.  Wood  Pulp  and  the  Textile  Industries. 
Bibliography.  Index. 


Photography:  its  Principles  and  Applications. 
By  ALFRED  WATKINS,  F.R.P.S.  342  pages.  98  Illus- 
trations. 

CONTENTS  :  First  Principles.  Lenses.  Exposure  Influences.  Prac- 
tical Exposure.  Development  Influences.  Practical  Develop- 
ment. Cameras  and  Dark  Room.  Orthochromatic  Photography. 
Printing  Processes.  Hand  Camera  Work.  Enlarging  and  Slide 
Making.  Colour  Photography.  General  Applications.  Record 
Applications,  Science  Applications.  Plate  Speed  Testing.  Pro- 
cess Work.  Addenda.  Index. 


IN  PREPARATION. 

Commercial  Paints  and  Painting.  By  A.  S.  JENN- 
INGS, Hon.  Consulting  Examiner,  City  and  Guilds  of 
London  Institute. 

Brewing  and  Distilling.    By  JAMES  GRANT,  F.S.C- 


I 


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Los  Angeles 

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